Transformations of Graphs and Digraphs

Western Michigan University ScholarWorks at WMU

Dissertations Graduate College

6-1991

Elzbieta B. Jarrett Western Michigan University

Follow this and additional works at: https://scholarworks.wmich.edu/dissertations

Part of the Applied Mathematics Commons

Recommended Citation Jarrett, Elzbieta B., "Transformations of Graphs and Digraphs" (1991). Dissertations. 2005. https://scholarworks.wmich.edu/dissertations/2005

This Dissertation-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Dissertations by an authorized administrator of ScholarWorks at WMU. For more information, please contact [email protected]. TRANSFORMATIONS OF GRAPHS AND DIGRAPHS

Elzbieta B. Jarrett

A Dissertation Submitted to the Faculty of The Graduate College in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Department of Mathematics and Statistics

Western Michigan University Kalamazoo, Michigan June 1991

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TRANSFORMATIONS OF GRAPHS AND DIGRAPHS

Elzbieta B. Jarrett, Ph.D.

Western Michigan University, 1991

Some distances defined on graphs depend on transforming one graph into

another. Two of these transformations are edge rotation and edge slide. In this

dissertation, extensions and generalizations of these transformations are investigated.

Chapter I begins with some preliminary definitions and known results. Then

two types of digraph transformations are introduced and their properties are studied.

Some measures of distance between graphs and distance between digraphs are

defined in Chapter II. Also distance graphs and digraphs associated with these

measures are introduced. Several known results concerning this topic are generalized

and new results are presented.

Chapter III is devoted to F-transformations, which is a generalization of the

previously discussed transformations of graphs. Based on F-transformations, a new

measure of distance between graphs and a new class of distance graphs (called F-

distance graphs) are introduced. A characterization of graphs that are F-distance

graphs is investigated.

Transformations of subgraphs and related topics are studied in Chapter IV.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INFORMATION TO USERS

This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer.

The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction.

In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion.

Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book.

Photographs included in the original manuscript have been reproduced

xerographically in this copy. Higher quality 6 " x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order.

UMI University Microlilms international A Bell & Howell Information Com pany 300 North Z eeb Road, Ann Arbor, tvtl 48106-1346 USA 313/761-4700 800/521-0600

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Order Number 9129602

Transformations of graphs and digraphs

Jarrett, Elzbieta Bozena, Ph.D.

Western Michigan University, 1991

UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS

I would like to express my deepest gratitude to Professor Gary Chartrand for

his guidance, assistance, and support throughout the entire period of my studies at

Western Michigan University, Kalamazoo, in particular during my work on the

dissertation. His encouragement and belief in me were especially helpful in

overcoming my moments of self-doubt. I truly feel honored to have been a student of

Professor Chartrand and to have had an opportunity to participate in research projects

conducted by him.

Special thanks to my first teacher of graph theory. Dr. Maciej M. Syslo, who

introduced me to graph theory and whose interesting and dynamic lectures inspired me

to make graph theory my career.

I would like to thank Professors Allen Schwenk and Arthur T. White for their

exceptional lectures, and for sharing their knowledge, research experience and

enthusiasm.

Many thanks to Professors Shashi F. Kapoor, Dalia Motzkin, Maciej M. Syslo

and Arthur T. White for serving on my committee.

I wish to express my gratitude to my dearest parents, Janina and Wladyslaw

Hubicki, for their unlimited love and support. I am especially grateful to my Mom,

who first showed me the beauty of mathematics and the fun of working in this field of

science.

Many tlianks to my son, Patrick, for his love and for understanding that a busy

mom does not mean an unloving mom.

I am also grateful to Tim for his love and support, especially during the last year

of my work. ii

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Finally, many thanks to Lisa Hansen, Rochelle Cullip, and Andrea Frey for

their assistance.

Elzbieta B. Jarrett

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS

ACKNOWLEDGEMENTS...... ii

CHAPTER

I . ROT ATION AND SLIDE TRANSFORMATIONS...... 1

1.1 Introduction ...... 1

1.2 Edge Rotation and Edge Slide Transformations ...... 2

1.3 Arc Rotation and Arc Slide Transformations ...... 4

II. DISTANCE GRAPHS AND DIGRAPHS...... 8

2.1 Distance Between Graphs...... 8

2.2 Arc Rotation and Arc Slide Distance ...... 12

2.3 Arc Rotation Distance Graphs and Arc Slide Distance D ig rap h s...... 17

III. F-TRANSFORMATIONS...... 35

3.1 A Generalization of Edge Rotation and Edge Slide Transformations ...... 35

3.2 Properties of F-Transformations ...... 37

3.3 F-Distance ...... 51

3.4 F-Distance Graphs...... 58

IV. TRANSFORMATIONS OF SUBGRAPHS...... 62

4.1 Edge Slide Subgraph Transformation ...... 62

4.2 Triangular Lihne Graphs...... 67

4.3 An Introduction to Subgraph Slide Distance Graphs ...... 79

4.4 Some Problems Concerning Subgraph Distance Graphs 83

REFERENCES...... 90

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER I

ROTATION AND SLIDE TRANSFORMATIONS

1.1 Introduction

Recently various measures of distance on classes of graphs have been explored.

Zelinka [13] defined the distance between two graphs Gj and G 2 of the same order p

as p - m, where m is the maximum order of a graph that is (isomorphic to) an induced

subgraph of both Gj and G 2 . Johnson [10] called this distance the mdMcecfjfM&grûp/j

metric and denoted it by dj(Gj, G 2 ). Zelinka [14] studied an analogue of this metric

for trees having the same order. Chartrand, Saba, and Zou [7] studied a distance

between graphs having the same order and same size. Their edge rotation distance

between graphs Gj and G 2 is based on the minimum number of edge rotations

required to transform Gj into G 2 . The three types of distance mentioned above were

compared by Zelinka in [15].

A metric on the space of all graphs has been defined by Johnson [9]. The

subgraph metric dg(G^, G 2 ) between graphs Gj and G 2 is based on the subgraph of

greatest cardinality (the sum of the order and the size of the subgraph) of both Gj and

G2 . Johnson [10] introduced a restricted version of the edge rotation distance, namely

theedge shift distance (referred to as tlie edge slide distance in [2], [3]) and explored a

partial ordering of metrics defined on the space of graphs.

Johnson [9] showed that the metrics defined on graphs may be applied to

problems in medicinal chemistry. A mathematical model of organic chemistry also

gives rise to this and is studied further by Bald%, Koba, Kvasnibka and Sekanina [1].

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Benadé, Goddard, McKee and Winter [2] explored three measures of distance

between graphs based on deformations which transform a graph Gj into a graph G2 .

These authors were concerned with the edge move distance, the edge rotation distance

and the edge slide distance, with the main focus on the last type of measure.

Faudree, Schelp, Lesniak, Gyarfas and Lehel [ 8 ] gave upper and lower bounds

on the (edge) rotation distance between two graphs in terms of their greatest common

subgraphs and their "partial rotation link" of greatest cardinality. Some extremal

problems for the (edge) rotation distance of trees were also proposed.

Chartrand, Goddard, Henning, Lesniak, Swart and Wall [3] introduced

distance graphs and investigated graphs that are distance graphs. An investigation of

the problem of characterizing distance graphs was continued in [ 8 ].

In this dissertation we study transformations of graphs and digraphs. In this

chapter we focus on edge and arc rotations, and edge and arc slides. In Chapter II we

investigate those graphs that are edge and/or arc rotation distance graphs. In Chapter

III we generalize the concept of graph transformation to F-transformations for a graph

F and study their properties. We also introduce F-distance graphs and investigate

graphs that are F-distance graphs. Transformations of subgraphs and measures of

distance between subgraphs are discussed in Chapter IV.

1.2 Edge Rotation and Edge Slide Transformations

In [2], [7] and [10] two types of transformations on graphs were defined. Let

G and H be two (p, q) graphs. We say that G can be transformed into H by an

edge rotation if G contains distinct vertices u, v and w such that uv g E(G), uw g

E(G) and H = G - uv + uw. More generally, we say that G can be r-transformed

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. into H if there exists a sequence G = Gq, Gj, G„ = G„ (n ^ 0) of graphs such

that Gj can be transformed into Gj^j by an edge rotation for i = 0 , 1 , n - 1 .

An edge slide is a restricted version of an edge rotation. A graph G can be

transformed into a graph H by an edge slide if G contains distinct vertices u, v and

w such that uv e E(G), vw e E(G), uw g E(G) and H = G - uv + uw. If a graph

H is isomorphic to a graph G or H can be obtained from a graph G by a sequence

of edge slides, we say that G can be s-transformed into H. For example, the graph

H of Figure 1.1 can be obtained from the graph G by an edge rotation, but H cannot

be obtained from G by an edge slide. On the other hand, the graph H' can be

obtained from G' by an edge slide (as well as by an edge rotation).

V w G: 9

V w G': O------O

Figure 1.1

It was shown in [7] that every (p, q) graph G can be r-transformed into any

other (p, q) graph H. It was also shown (in [10]) that s-transformation preserves

connectedness. Therefore, a graph G can be s-transformed into a graph H if and

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. only if G and H have the same number of components and corresponding

components of G and H have the same order and same size.

In this dissertation we extend the above two transformations to digraphs.

1.3. Arc Rotation and Arc Slide Transformations

Let D and F be two digraphs having the same order and same size. We say

that D can be transformed into F by an arc rotation if D contains distinct vertices u,

V and w such that (u, v) € E(D), (u, w) g E(D), and F = D - (u, v) + (u, w). For

example, the digraph F of Figure 1.2 is obtained from the digraph D by an arc

rotation.

V w V w D: ° F:

Figure 1.2

If a digraph D can be transformed into a digraph F by an arc rotation, then

clearly F can be transformed into D by an arc rotation. Two such digraphs D and F

are called ar-adjacent. The digraph D is said to be ar-transformed into F if there exists a sequence Fq, Fj, ..., Fj, (n > 0) of digraphs such that Fg = D, F^ = F and

Fj_j is ar-adjacent to Fj, for i = 1, 2, ..., n. The relation "can be ar-transformed

into," which we denote by R, is an equivalence relation on the set of digraphs.

Moreover, if D and F are digraphs that belong to the same equivalence class of R,

then D and F have the same order and same size. However, the converse is not true.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in general. For example, the digraphs D and F of Figure 1.3 have the same order

and same size, but neither one can be ar-transformed into the other.

Cb— ^ -----O p . Q------►— O D: V ■ V

Figure 1.3

Observe that in the process of transforming one digraph into another by an arc

rotation, the outdegree of each vertex is preserved. Therefore, if D can be ar-

transformed into F, then D and F have the same outdegree sequence. We show that

the converse of this implication is also true.

Proposition 1.1 If D and F are two digraphs having the same outdegree

sequence, then D can be ar-transformed into F.

Proof If D = F, then, by definition, D can be ar-transformed into F; so suppose

that D ^ F . Without loss of generality, assume that D and F have the same vertex set,

namely V(D) = V(F) = {vi, v%, ..., Vp}, where odo v; = odp v, for i = 1, 2,..., p.

Suppose, to the contrary, that D cannot be ar-transformed into F. Among all the

digraphs into which D can be ar-transformed, let F' be one having a maximum

number of arcs in common with F. Since F' and F have the same vertex set, the

same outdegree sequence (with odp vi = odp' vi, i = 1, 2,..., p), and F' é F, there

exist two arcs (v,, vj) and (vj, Vj.) with the property that (vp Vj) e E(F') - E(F) and

(vi, Vk) e E(F) - E(F'). Let F" denote the digraph obtained from F' by rotating an

arc (vp Vj) into (vp v,^). Then D can be ar-transformed into F" (by transitivity of

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6

ar-transformations) and F" has more arcs in common w ith F than F' does. This

contradicts the choice of F and yields the desired result that F can be obtained from

D by ar-transformation. □

The transformations of graphs and digraphs discussed thus far are symmetric.

Now we introduce a transformation (defined on digraphs) that lacks symmetry. Let D and F be two digraphs having the same order and same size. We say

that D can be transformed into F by an arc slide if D contains distinct vertices u, v

and w such that (u, v) e E(D), (u, w) <£ E(D), (v, w) e E(E>), and F = D - (u, v) +

(u, w). As mentioned earlier, arc slide transformations are not symmetric. For

example, the digraph D of Figure 1.4 can be transformed into F by an arc slide,

while F cannot be transformed into D by an arc slide.

n ' F:

Figure 1.4

We say that a digraph D can be as-transformed into a digraph F (or F can be as-transformed from D) if there exists a sequence Fq, F j, F„ of digraphs such

that D =F q, F = F^ , and Fj can be obtained from F j _ j by an arc slide for i = 1, 2 ,

..., n. Since an arc slide is a restricted version of an arc ro tatio n , the fact that D can be

as-transformed into F implies that D can be also ar-transform ed into F, and therefore D

and F must have the same outdegree sequence. Even though the digraphs D and F

of Figure 1.4 have the same outdegree sequence, F cannot b e as-transformed into D.

In fact, F cannot be as-transformed into any digraph different from F.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Note that if D is a connected digraph and F is a digraph obtained from D by

an arc slide, then F is connected. Therefore, for as-transformations we restrict our

study to connected digraphs only.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTERu

DISTANCE GRAPHS AND DIGRAPHS

2.1 Distance Between Graphs

In Section 1.2 two types of transformations on graphs were discussed.

Associated with these transformations are two metrics defined on graphs (see [7],

[10]). Let G and H be two graphs having the same order and same size. The edge

rotation distance or, more simply, the r-distance d^(G, H) between G and H is the

smallest nonnegative integer n for which there exists a sequence G q , Gj,..., G„ of

graphs such that G =G q , H = G „ and Gj can be obtained from Gj_j by an edge

rotation for i = 1, 2,..., n. For example, the edge rotation distance between graphs G

and H shown in Figure 2.1 is d^(G, H) = 3.

G; A O H; O-

Figure 2.1

The following properties of edge rotation distance were established in [7]. The

complement of a graph G is denoted by G.

Proposition 2A If G and H are two graphs having the same order and same size,

then dj.(G, H) = d^G, H) .

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It was shown that every nonnegative integer is the r-distance between some

pair of graphs.

Proposition 2B For every nonnegative integer n, there exist graphs G and H

such that dj(G, H) = n.

Prior to presenting an upper bound for the r-distance between two graphs, we

introduce another concept. For nonempty graphs G^ and G 2 , a greatest common

subgraph of Gj and G 2 is defined as any graph G of maximum size without

isolated vertices that is a subgraph of both Gj and G 2 .

Proposition 2C Let G and H be two (p, q) graphs with q > 1, and let s be the

size of a greatest common subgraph of G and H. Then d^(G, H) < 2(q - s).

Moreover, the above bound is sharp.

Another concept of a distance between graphs is associated with edge slide and

was discussed in [2] and [10]. Let G be a graph with components Gj, 1 < i < k, and

H a graph with components Hj, 1 < i < k, such that Gj and Hj have the same order

and same size. We define the edge slide distance or, simply, s-distance dg(G, H)

between G and H as the smallest nonnegative integer n for which there exists a

sequence G = G q , G j, ..., G^ s H of graphs such that, for i = 1, 2 n, Gj can

be obtained from Gj_j by an edge slide. If G and H are the graphs presented in

Figure 2.2, then the edge slide distance between G and H is dg(G, H) = 2.

Note that d^(G, H) = 1 for the graphs G and H of Figure 2.2. It is

straightforward to show that d^(G, H) < dg(G, H) for every pair G, H of connected

graphs having the same order and same size. The following result is perhaps less

obvious.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10

G: H:

Figure 2.2

Proposition 2.1 For every pair m, n of positive integers with m ^ n, there exist graphs G and H such that d^(G, H) = m and dg(G, H) = n.

Proof Let P: Vg, Vj,Vj,+ni be a path on n + m + 1 vertices. Define G as the

graph obtained from P by adding m new vertices Wj,W2 , ..., w ^ and 2m edges,

namely WjV2 i_ 2 and WjV 2 j, for i = 1, 2 , ..., m - 1 , and w,^V 2 m_ 2 and

We also define H as the graph obtained from P by adding m vertices Wj, W 2 , ...,

w^ and joining every vertex wj only with V 2 ;_% and V 2 j, if 1 < i < m - 1 , and with

''n+m - 1 v„^^ if i = m. For m = 3 and n = 4, the graphs G and H are shown

in Figure 2.3.

Observe that the graph G does not contain vertices of degree 1 or 3, while H

has 2m such vertices (one vertex having degree 1 and 2m-1 vertices of degree 3).

Since one edge rotation changes the degree of at most two vertices, d^(G, H) > m.

Now let G q , G j , ..., Gjj, be graphs defined as follows: G q = G, G, = Gj_j -

WiV2 i_ 2 + WiV2 i_j, for i = 1 , 2 ,..., m - 1 , and G^, = G ^.i - WmV 2 m_ 2 +

Note that G^ = H and, for i = 1, 2,..., m, the graph Gj is obtained from Gj_j by

an edge rotation. Thus we have d^(G, H) = m.

Observe also that, for i = 1, 2,..., m - 1, the graph Gj can be obtained from

Gj_j by an edge slide. Define graphs Hg, H j , ..., as follows: Hg = G^_j

and, for i = 0, 1,..., n + m, the graph Hj+j = Hj - w^V2 m_2 +i + w^V2 m_i+i. Since

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11

for i = 0 , 1 , n + m, the graph Hj can be transformed into Hj^j by an edge slide,

and - H, the graph G can be s-transformed into H and we have dg(G, H) <

(m - 1 ) + (n - m -t- 1) = n. On the other hand, G contains m - 1 4-cycles and one

(n - m + 4)-cycle (where n - m + 4 > 4), while H has m 3-cycles. Since cycles in

G (and in H) are edge disjoint, one edge slide may decrease (by at most 1) the length of only one cycle. Therefore, dg(G, H) ^ (m - 1) • 1 + (n - m -t- 1) = n and the

desired result follows. □

H: V

Figure 2.3

For the two metrics on graphs discussed above, the corresponding distance

graphs were introduced in [3]. Let S be a set of (nonisomorphic) (p, q) graphs. Then we define the edge rotation distance graph (D^{S) of S as the graph with the

vertex set S such that two vertices G and H of (D^(S) are adjacent if and only if

d/G, H) = 1.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12

Let S' be a set of (nonisomorphic) graphs having the same number of

components, labeled in such a way that the ith components of all graphs have the same

order and same size. Then we define the edge slide distance graph ®g(S') of S'

analogously.

It was shown in [3] that every graph is an edge slide distance graph and it was

conjectured that all graphs are edge rotation distance graphs. A number of classes of

graphs are known to be edge rotation distance graphs.

Proposition 2D ([3]) Complete graphs, cycles and trees are edge rotation distance

graphs.

Proposition 2E ([3]) Every line graph is an edge rotation distance graph.

Proposition 2F ([ 8 ]) The complete bipartite graphs, K3 3 and K 2 p (p ^ 1) are

edge rotation distance graphs.

2.2 Arc Rotation Distance and Arc Slide Distance

For two digraphs D and F having the same outdegree sequence we define the

arc rotation distance (or ar-distance) d^(D, F) between D and F as the smallest

nonnegative integer n for which there exists a sequence Dq, D j, ..., D^ of digraphs

such that D = Dg, F = D^, and Dj and Dj^j are ar-adjacent for i = 0, 1,..., n - 1.

By Proposition 1.1 this distance is a well-defined concept. Moreover, if ^s) is the

set of all digraphs having outdegree sequence s, then (f(s), d^j.) is a metric space.

Arc rotation distance and edge rotation distance have many similar properties, as

we now illustrate.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13

Proposition 2.2 If D and F are two digraphs having the same outdegree sequence, then dg^(D, F) = d^D , F).

Proof If dgj.(D, F) = 0, then D = F. Therefore D = F and dgj(D, F) = 0. Assume,

then, that dg^(D, F) = n > 0. By the definition of the arc rotation distance, there exists

a sequence D = Dq, D j, ..., D^^ = F of digraphs such that Dj can be transformed into

Dj+j by an arc rotation for i = 0,1,..., n - 1. Observe that if the digraph Dj can be

transformed into Dj^j by an arc rotation, that is, Dj^j = Dj - (vj, Vj) + (vj, Vj^), then

the digraph Dj can be transformed into Dj^j by an arc rotation since Dj^j = Dj -

(vj, Vj^) + (Vj, Vj). Thus the existence of the sequence D = Dq , D j , ..., D^ = F implies

that dgj(D, F) < n = d^(D, F). Applying the same argument to D and F, we obtain

dg^D, f ) < dgj(D, F). A s a consequence, d^^(D, F) = d^/D, F). □

The following result is completely analogous to a result on graphs (see [3]).

Proposition 2.3 If Dj and D 2 are two digraphs having the same outdegree

sequence and D[ (i = 1,2) is the digraph obtained from Dj by adding a new vertex

adjacent to all vertices of Dj, then dgj.(Dj, D 2 ) = 1 if and only if dgj.(D{, D^) = 1.

Proof Clearly, if d^^(Dj, D 2 ) = 1, then dg^(Dj, D^) = 1. For the converse, suppose

that dgj.(Dj, D 2 ) = 1. Without loss of generality, assume that Dj and D 2 have the

same vertex set, namely V(Dj) = V(D 2 ) = V = (vj, V2 , ..., Vp), and V(D|) = V(Dj) =

Vu{Vp^j}. Since Dj can be transformed into D 2 by an arc rotation, there exist arcs

(vj, Vj) and (vj, Vj.) such that (vj, Vj) e E(D j) - ECD^), (vj, Vj^) e E(D^) - E(D j)

and D 2 = D j - (vj, Vj) + (vj, Vj^). Note that Vj Vp+j since both digraphs Dj and

D2 contain all arcs (Vp^j, v^), for m = 1, 2,..., p. Moreover, since the indegree of

Vp+j is 0 in both Dj and D^, we have Vj Vp^i and Vj^^tVp^j. Therefore,

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14

(vj, Vj) e E(Dj) - E(D2 ) and (vj, v^> e E(D 2 > - E(D)), and D 2 = Dj - (vj, Vj) +

(vj, Vk). Thus, d^(D j, D 2 ) = 1. □

Corollary If Dj and D 2 are two digraphs having the same outdegree sequence and

D- (i = 1, 2) denotes the digraph obtained from Dj by adding a new vertex adjacent to

all the vertices of Dj, then d^^CD^, D 2 ) = dg^.(Dj, D^).

We next define a distance related to arc slide. For two digraphs D and F such

that D can be s-transformed into F, we say that the arc slide distance (or as-distance) dgg(D, F) from D to F is the smallest nonnegative integer n for which there exists a

sequence Dq, D^, ..., D^ of digraphs such that D = Dq, F = and can be

obtained from Dj by an arc slide, for i = 0, 1,..., n - 1. Note that there exist pairs

D, F of digraphs such that D cannot be as-transformed into F. For such pairs the

arc slide distance is defined to be infinity. Recall that all previously defined distances

are metrics, so they have the symmetric property. This property does not hold,

however, for arc slide distance. For example, if D and F are digraphs of Figure 2.4, then dgg(D, F) = 1 while dgg(F, D) = 2.

D: F:

Figure 2.4

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15

Although as-distance is not symmetric, it satisfies the other two properties of a

metric, that is, (1) dgg(D, F)=0 if and only if D = F, and (2) for digraphs D, F and

H having the same outdegree sequence, d^g(D, F) + d^g(F, H) > dgg(D, H). Observe

also, that for every pair D, F of digraphs having the same outdegree sequence,

d^(D, F)

next.

Proposition 2.4 For every pair m, n of positive integers with m < n, there exist digraphs D and F such that dg^(D, F) = m and dgg(D, F) = n.

Proof Let P: v = V q , Vj , ..., v ^ be a directed path and let C: u = Ug, U j, ...,

^n-m+ 3 ’ "o ^ directed cycle. Define D as the digraph obtained by identifying the

vertex v of P with the vertex u of C. Now, let P' be a directed path on n -m + 2

vertices, let S be a star of order m + 1 with all arcs directed towards the center, and

let C': X, y, z, x be a directed 3-cycle. Then we define F to be the digraph obtained

from P', C' and S by identifying two pairs of vertices, namely, the vertex r of P'

having outdegree 0 with the vertex x of C', and the vertex y of C' with an end-

vertex t of S. For instance, if m = 3 and n = 5, the digraphs D and F are shown

in Figure 2.5.

Observe that the vertex of outdegree 0 has indegree 1 in D, while the vertex

of outdegree 0 has indegree m in F. Moreover, D contains a (n - m + 4)-cycle,

while the only cycle of F has length 3. Thus we have the following inequalities

dgg(D, F) ^ (m - 1) + (n - m + 4 - 3) = n and dg^(D, F) > (m - 1) 4 1 = m.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16

D: "4

u , = V, u 3

u u 2 1

r =

Figure 2.5

On the other hand, there exists a sequence D =D q, D^, = F of digraphs, such

that, for i = 1 , 2 ,..., n, the digraph Dj can be obtained from by an arc slide.

Namely,

Dq = D,

D; = Dj_i - (uq, Uj) + (uq, Uj^j), 1 < i < n - m + 1 ,

Di = Di_i - (Vn_i, + (v„_i, v^), n - m + 2 < i< n .

Thus dgg(D, F) < n. Since the digraph can be obtained from Dq by an arc

rotation (namely, = Dq - (uq , Uj) + (uq , Un-m+l))» have d^j(D, F) < m

and the desired result follows. □

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17

2.3 Arc Rotation Distance Graphs and Arc Slide Distance Digraphs

Let S be a set of (nonisomorphic) digraphs having the same outdegree

sequence. Then thearc rotation distance graph D^(S) is defined to be the graph with

vertex set S such that two vertices Dj and D 2 of ^ ^ (S ) are adjacent if and only if

the digraphs D^ and D 2 are ar-adjacent.

Analogously, the arc slide distance digraph (D^(S) of S is the digraph having

S as its vertex set, with a vertex Dj of 2^g(S) adjacent to a vertex D 2 if and only if

the digraph Dj can be transformed into D 2 by an arc slide.

We investigate here two questions: "Which graphs are arc rotation distance

graphs?" and "Which digraphs are arc slide distance digraphs?" We present a number

of families of graphs that are arc rotation distance graphs, and we prove that all

digraphs are arc slide distance digraphs. We begin with the second question.

Proposition 2.5 Every digraph is an arc slide distance digraph.

Proof Let D be an arbitrary digraph with vertex set V(D) = {vj, V 2 , ..., Vp}, and

let F be the digraph obtained from D by adding, for each Vj (1 ^ i < p), a total of 2i p new vertices, each adjacent only to Vj. Thus, F has order p + ^ 2i = p + 2(P 2 ) • i=l Next, for i = 1, 2, ..., p, we define Fj as the digraph obtained from F by adding

another new vertex adjacent only to Vj. Each digraph Fj is characterized by the

sequence ti: tj, t^, ..., t^, where tj ( 1 < j < p) denotes the number of vertices in Fj

having indegree 0 and adjacent to the vertex Vj. For 1 ^ i < j < p, we have

t\ 2, 4, ..., 2i - 2, 2i + 1, 2i + 2,..., 2j + 2, 2j, 2j + 2, ..., 2p, and

th 2, 4 ,..., 2i - 2, 2i, 2i + 2,..., 2j - 2, 2j + 1, 2j + 2...... 2p.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18

Thus dgg(F|, Fj) = 1 if and only if in Fj we can "slide" the end-vertex Vj of an arc

(x, Vj) (where x is a vertex of indegree 0 adjacent to Vj in Fj) to the vertex Vj, that

is, if and only if Vj is adjacent to Vj in D. □

The above construction is based on the construction used in [ 8 ] to prove that

every graph is an edge slide distance graph. Next we investigate graphs that are arc

rotation distance graphs. First we show that complete graphs are arc rotation distance

graphs.

Proposition 2.6 For p > 1, Kp is an arc rotation distance graph.

Proof Let P: Vg, V j,..., Vp ^ 3 be a directed path. For i = 1, 2,..., p, define Dj to

be the digraph obtained from P by identifying the end-vertex Vp^g with Vj (as

illustrated in Figure 2.6 for p = 4). Since dg^(D;, Dj) = 1 for 1 < i j < p, we have

2)^({Di,D2, ...,Dp}) = Kp. □

Di=

V V 6 6

V,6

Figure 2.6

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19

Next, we show that stars are arc rotation distance graphs.

Proposition 2.7 For n > 2, the star Kj ^ is an arc rotation distance graph.

Proof Let C^* ^ V 2 , v„^j, denote a symmetric (n + l)-cycle and let

be the digraph obtained from by adding, for each j = 1 , 2 , ..., n, a total

of 2j new vertices, each adjacent only to Vj. Then for i = 1, 2, ..., n, let

Dn+i = D n + i- (x, Vj) + (x, where x is a vertex in having indegree 0 .

This construction is illustrated for n = 3 in Figures 2.7 and 2.8. Digraphs C 4 and

D4 are shown in Figure 2.7(a) and the corresponding simplified drawings are

presented in Figure 2.7(b). In Figure 2.8 the simplified drawing is used to illustrate

digraphs D 4 , d |, and D 4 .

(b)

Figure 2.7

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20

Figure 2.8

First observe that, by the construction of we have j, = 1

for i = 1, 2 , p. It remains to show that, for 1 < i < j < p, the digraphs Dj,^j and

are not ar-adjacent. Let tj (1 < i < n, 1 < j < n + 1) denote the number of

vertices of having indegree 0 (and called 0-indegree vertices) that are adjacent

to the vertex vj. If t* (1

t‘: 2, 4 ,..., 21 - 2, 2i - 1, 2i + 3, 2i + 4, ..., 2(n + 1). Note that, for 1 < i < j ^ n, the

sequences t' and t^ differ in at least three positions. Since every arc rotation on

changes exactly two values of t', the digraph Djj|j cannot be obtained from by

an arc rotation. Hence ®gj.({Do, D j,..., D„}) = □

The digraphs defined in the proof of the previous theorem are very useful for

proving a more general result which we state next.

Proposition 2.8 Every tree is an arc rotation distance graph.

Proof Let T be a tree. We may assume that T is rooted at a vertex v having

degree A(T) = A. Denote the height of T by h(T) and label the vertices of T

(recursively) as follows. First label v as 1. Then label the k children of 1 as 1,1;

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21

1, 2; 1, k. In general, label the kj children of vertex t by t, 1; t, 2; t, k^.

Note that the label of every vertex (except the root) contains information about the label

of its parent and about the position of a vertex among its siblings. For instance, the

vertex 1,3,2 is the second child of the vertex 1,3. An example of a tree labeled in

such a way is shown in Figure 2.9.

1, 1,1 1, 1,2 1,3,1

Figure 2.9

Now we construct (recursively) digraphs corresponding to the vertices of a

given tree T and show that two of these digraphs are adjacent if and only if the

corresponding vertices of T are adjacent. Each digraph consists of h(t) components. The digraph corresponding to a vertex t of T is denoted by F^ and its components

by F{, 1 < i < h(T).

First we define the components of Fj (the digraph corresponding to the root of

T) by

i = 1, 2, ..., h(T),

where Dj is the digraph defined in the proof of Proposition 2.7. For example, for the

tree T of Figure 2.8, h(T) = 2, A = 3 and the two components of Fj are Fj = D®

and F j = Dg, as shown in Figure 2.10.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22

Figure 2.10

Suppose that we have already defined the digraphs corresponding to the vertices

of T that are at distance d from the root, and let t be a vertex of T whose distance

from the root is d. ITien the components of the digraph corresponding to the vertex

t,j (the jth child of t) are p jj F g where

F‘t 1 < i < d

1 A+d+l i = d +

\P\ d + 1< i < h(T)

In other words Fg = f J u F^ u ... u F^ u u u ... u For

the tree T of Figure 2.9, the seven digraphs corresponding to the vertices of T are

shown in Figure 2.11.

We show that dg^.(F^, Fg) = 1 if and only if t and s are adjacent. Let t and s

be two adjacent vertices of T, that is, one is the parent of the other. Without loss of

generality, assume that t is the parent of s. Therefore, s = t, j for some j > 0. Observe that the corresponding digraphs F^ and Fg differ in exactly one component.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23

namely the (d + 1) st (where d is the distance from the root to the vertex t in T).

Moreover,

A+d+l ’

so can be obtained from by an arc rotation (see the proof of Proposition

2.8). Hence, F^ and Fg are adjacent.

For the converse, the following observations will be useful.

Observation 1 Each digraph has two types of arcs, namely, free arcs

(incident/rom a vertex of indegree 0 ) and cyclic arcs belonging to the subdigraph of

isomorphic to the symmetric directed cycle C^. If one of the digraphs F^ and Fg

can be obtained from the other by an arc rotation, say Fg = F^ - (x, y) + (x, z), then

(x, y) and (x, z) are free arcs.

Observation 2 The ith component of every digraph F^ is determined by the

presence of a subdigraph isomorphic to Moreover, all ith components have the

same order (that is, p(F{) = p(Fg) for all t, s 6 V(T)). This fact and Observation 1

imply that if F^ can be transformed into Fg by an arc rotation, say Fg = F^ - (x, y) +

(x, z), then the rotation is performed within one component, that is, (x, y) e F| and

(x, z) 6 Fg for some 1 < i < h(T), (for if (x, y) e F[, ( x , z) 6 F^g and i j, then

p(F;) = p(F'g) + l>p(F;)).

Observation 3 If t is a vertex of T at distance d from the root, then

V U ... U U ^ ^ ^A+h(T)’ ij > 0, 1 ^ j ^ d.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24

" v y ^ .

d7 ^ ■" v/<

d f ^

d T ^ d f

,3,r

d f l D

Figure 2.11

Assume now that s and t are two nonadjacent vertices of T. Let dg and d^

denote the distance from the root to s and to t, respectively. Without loss of generality, assume that dg > d^. We consider three cases.

Case 1. Assume that dg = d^ = d. Then we have

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25

F, = ^ ^ ^ ^

and

Fs = Dj>,, u u ... u u D » ,,,, u ... u D % (T ).

with > 0 and jj^ > 0 , 1 < k < d.

Since Fg, there exists an integer k (1 < k < d) such that ï* ^ + k ’

ijj jjç^. But then, as shown in the proof of Proposition 2.8, cannot be

transformed into lYk and therefore F, and F„ are not adjacent. A+k '■ "

Case 2. Assume that dg = d^ + 1. Since t and s are not adjacent, there exists a

vertex t' (distinct from t) at distance d^ from the root which is the parent of s. As

shown in Case 1, F^ and F^, differ in at least one, say the kth, component, that is,

F[ ^ F{i. Moreover, neither F^ nor F|^ can be transformed into the other by an arc

rotation. Note that F^ = FjS and therefore Fj and Fg are not adjacent.

Case 3. Assume that dg>d^+l. Then F^ and Fg differ in at least two components

since

andF^s = D L ^

while

for some j > 0 and k > 0. Hence F^ and Fg are not adjacent. □

Another two families of graphs that are arc rotation distance graphs are cycles

and wheels.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26

Proposition 2.9 For n ^ 3, is an arc rotation distance graph.

Proof Let P: Vj, V2 ,v„, be a directed path and let D be the digraph

obtained from P by adding two new vertices v ^_ ^2 and v^^_^g and three new arcs

K + l ’ Vy+2 ), (Vn+2 , v„+3 ) and (v^+g, Then, for i = 1, 2 , n - 1, we define

Dj to be the digraph obtained from D by adding two other vertices x and y such that

X is adjacent only to Vj and y is adjacent only to Vj^j. We also define Dj^ as the

digraph obtained from D by adding two new vertices, namely vertex x adjacent only

to Vj, and vertex y adjacent only to Vj. For n = 4, the digraphs D j, D 2 , Dg and D 4

are shown in Figure 2.12.

^ 1 ’ ^ 2 ' X y Vg v.y X y Vg v.^

^ 2 ""a \ ""s ''i ""2 ""s "^4 ^ 5

D3 : D ,:

X y Vg v.^ y X Vg v ^ _ i X F i ™ L F ^2 "^3 ^^4 ''S "'l "^2 "^3 \ ^5

Figure 2.12

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27

In every digraph Dj (1 ^ i ^ n) the vertices Vj, V2 , v^^g are uniquely

characterized by their distance from Vj. Namely, for each j (1 < j < n + 3), the vertex

Vj is the only vertex whose distance from Vj is j - 1. Moreover, for 1 < i < j < n

and j i + 1 , the four vertices Vj, Vj^j, Vj and Vj^j have different indegrees in the

digraphs Dj and Dj. Hence Dj cannot be transformed into Dj by an arc rotation and

therefore d^(Dj, Dj) > 1 (for 1 < i < j < n, j i + 1). On the other hand, for i = 1,

2 , ..., n - 1 , the digraph Dj+j = Dj - (x, Vj) + (x, V;+2 ): so d ^/D ;, D j^i) = 1 .

Similarly, dgj.(Di,Dn)= 1 and ©^({Dj, D 2 , ..., D„}) = C^. □

Let C: Uj, U2 , ..., u^, Uj be a directed cycle. Define Dq to be a digraph

consisting of two components Dqj and Dq 2 - The first component is obtained from C

by adding two new vertices x and y such that x is adjacent only to Uj and y is

adjacent only to U 2 - We define Dq 2 to be the directed 3-cycle "»+!' "n+ 2 ' "n+ 3 '

u„^j. For instance, if n = 4, the digraph Dq is shown in Figure 2.13. Now, if

D j, D 2 , ..., Djj are the digraphs defined in the proof of Proposition 2.9, then, for

i = 1, 2, ..., n, the digraph Dq = Dj - (v„, v„+j) + (v„, Vj) and d^CDg, Dj) = 1. As

an immediate consequence, we have the following. (We write Wj „ for the wheel

Kj + Cjj.)

Proposition 2.10 For n > 3, the wheel Wj ^ is an arc rotation distance graph.

By a slight modification of the digraphs used in proofs of the last two

propositions, we show that complete bipartite graphs are arc rotation distance graphs.

Proposition 2.11 For n > m > 3, the complete bipartite graph is an arc

rotation distance graph.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28

Dq : ^ y

u 1

u "4 3 5 "7 D ^ 0 1

Figure 2.13

Proof Let P: Vg, Vj,V2 n+m ^ directed path, and let C: Uj, U 2 , U3 , Uj be a

directed cycle. Define D to be the digraph obtained by identifying the vertex V 2 „+^

of P with the vertex Uj of C. Then, for 1 < i < n, define Dj to be the digraph

obtained from D by adding two new vertices x and y such that x is adjacent only to

V2 i_ 2 and y is adjacent only to V 2 j (both vertices x and y have indegree 0). For

n = 4 and m = 3, the digraphs D j,D 2 , D 3 and D 4 are shown in Figure 2.14.

The digraphs Dj, D2 , ..., have the same order (namely, 2n + m + 5),

same size (also 2n + m + 5), and same outdegree sequence. Observe that in each

digraph Dj (where 1 < i < n), the vertices of indegree 2 are characterized by their

distance from Vq. Moreover, for i / j, two vertices of Dj having indegree 2

(namely, V2 j_i and V 2 ;) have indegree 1 in Dj, while the vertices Vjj,] and Vj of

indegree 2 in Dj have indegree 1 in Dj. Since an arc rotation changes indegrees of

exactly two vertices, Dj cannot be obtained from Dj by an arc rotation.

Next we define m digraphs F^, F 2 , ..., F,^ such that d^(Fj, Fj) > 1, for 1 <

i < j < m, and dg^(Dj, Fj) = l for all i and j with l^ i< n and l< j^ m . Then for

the set S = {Fj, F2 , ..., F ^, D j, D2 , .... D„), we have 2?^(S) = „.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29

Dy X y ^12 ^13

'■o '■l ''2 ''3 ''4 ''5 ''6 ''7 ''8 ''9 ''lO ''ll

Dg: X y ' ' 1 2 ' '1 3

^ 0 "^2 ^3 "^4 "^5 ^6 "^7 ""g \ ""lO ''ll

Dg: X y " 1 2 " 1 3 __LL_F "0 "1 "2 "3 "4 "5 \ "7 "g "9 "10 "11

X y " 1 2 " 1 3 __LL_F "0 "1 "2 "3 "4 "5 \ "7 "g "9 ""lo "'ll

Figure 2.14

For 1 ^ i < m, let Fj be a digraph containing two components, namely Fjj

and Fj 2 - The component F^ is obtained from the directed cycle C: Uj, U 2 , ...,

U2 n+i» U] by adding two new vertices x and y such that x is adjacent only to Uj

and y is adjacent only to U 2 - The component Fj 2 is obtained from the directed path

P': Wq, Wj,w^_; by adding two vertices and w^_j^2’ three arcs

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30

'Vm_i+2 ). and (w„_i+ 2 -For example, if n = 4 and

m = 3, we have the digraphs F1 .F 2 .F 3 of Figure 2.15.

X y W3 W4

Wg w^ w^

X y

W3 ‘22

w. w 0 "1

" 7 "6

X y F 3 :

u 10 F w

Figure 2.15

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31

The digraphs Fj, F2 , F^ have the same order and same size, namely,

I V(Fj) I = I E(Fj) 1 = 2n + m + 5 for 1 < i < m, and the same outdegree sequence. Suppose that for some i and j (1 < i < j < m), Fj can be obtained from Fj by an arc

rotation, that is, Fj s Fj - e + e' (for some e e E(Fj) - E(Fj) and e' e E(Fj) - E(Fj)).

Note that the components F^ and Fjj (i j) differ in size and in lengths of their

cycles. Since the arc rotation Fj - e + e' changes the length of the cycle of Fjj, the arc

e must be an arc of the cycle Uj, U2 U2 ^+i, Uj. Moreover, since this arc rotation

changes the size of F^, it cannot be performed within the component Fjj. Therefore,

e' is an arc joining a vertex u,. of Fjj and a vertex Wj of Fj 2 (for some 1

2n + i and 0 < t < m - i + 2) and the digraph F' = Fj - e + e' is connected.

However, the digraph Fj is not connected; thus it cannot be obtained from Fj by an

arc rotation and therefore we have d^^(Fj, Fj) > 1.

On the other hand, for 1 < i < n and 1 < j < m, the distance d^(Dj, Fj) = 1,

since Fj = Dj - (V2 n+j_j, V2 „+j) + (v2 n+j_j, Vq). Hence (D^({Fj, F 2 , ..., F^, D j, D 2 ,

= ^m,n' ^

Suppose now that S = {Dj, D 2 , ..., D^} is a set of nonisomorphic digraphs

having the same outdegree sequence. For i = 1, 2,..., n, let Gj be the underlying

graph of Dj. One can ask the question: How is the edge rotation distance graph

l?j({G j, G 2 , ..., G„)) related to the arc roation distance graph 2)g/(D j, ..., D^))?

First note that if some of the digraphs Dj (i = 1, 2,..., n) are not asymmetric, then the

edge rotation distance d^(Gj, Gj) may not be defined for some 1 ^ i, j < n. For

instance, if n = 2 and Dj and D 2 are the two digraphs shown in Figure 2.16, then

dar(Dj, D 2 ) = 1 but d^(Gj, G 2 ) is not defined since Gj and G 2 do not have the

same size.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32

D r

G r

Figure 2.16

Suppose that the digraphs Dj, D2 , a r e asymmetric. Then, necessarily,

the graphs Gj, G2 , ..., have the same order and same size, and dj.(Gj, Gj) is

defined for all 1 < i, j < n. In this case both distance graphs D ^ ({ D j,..., D^)) and

..., G^)) are defined. However, they do not necessarily have the same

order. For example, if n = 2 and Dj and D 2 are the two digraphs shown in Figure

2.17, then Gj = G 2 and the edge rotation distance graph (D^dG^, G 2 D has order 1

while the arc rotation distance graph D 2 )) has order 2 .

D D • Gj = G2 :

-O O

Figure 2.17

Let Dj, D 2 , ..., D^, be asymmetric digraphs having the same outdegree

sequence and such that their underlying graphs Gj, G 2 , ..., G„ are pairwise

nonisomorphic. If d^^fDj, Dj) = 1 for some 1 < i < j < n, that is, if there exist distinct

vertices x, y and z of Dj such that (x, y) e E(Dj), (x, z) g E(Dj) and Dj = Dj -

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33

(x, y) + (x, z), then we have Gj = Gj - xy + xz (with xy e E(Gj) and xz g E(Gj))

and dyCGj, Gj) = 1. This implies that, in general, dj.(Gj, Gj) < d^(D^, D j). To show

that this inequality can be strict, consider the digraphs D^, D 2 and Dg of Figure 2 .1 8 ,

where dg^(D^, Dg) = 2 while d^(Gj, Gg) = 1.

G,: 0 3 : V

Figure 2 .1 8

In general, the arc rotation distance between two nonisomorphic, asymmetric

digraphs having the same outdegree sequence is not equal to the edge rotation distance

between the corresponding underlying graphs. Suppose that D q , D j , ..., Dj^ are the

digraphs defined in the proofs of Propositions 2 .9 and 2 .1 0 and, for i = 0 , 1 , ..., n,

Gj denotes the underlying graph of Dj. Then dg,.(Dj, D j ) = 1 if and only if

dj(Gj, Gj) = 1 (1 ^ i < j < n). An immediate consequence of this is the following.

Corollary 2.10a For n > 3, the wheel Wj ^ is an edge rotation distance graph.

Similarly, for 1 < i < n and 1 < j < m, let Gj and Hj denote the underlying

graphs of Dj and Fj (the digraphs defined in the proof of Proposition 2 .1 1 ) ,

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34

respectively. Then dg^.(D|, D j) = 1 if and only if dj(Gj, G j) = 1, while d^(Fj, F j) = 1

if and only if dj.(Hj, Hj) = 1, and dgj(Dj, Fj) = 1 if and only if d^(G;, H j) = 1. This

implies the following result.

Corollary 2.11a For n > m > 3, the complete bipartite graph is an edge

rotation distance graph.

We know of no example of a graph which is not an arc rotation distance graph

and we conjecture the following.

Conjecture Every graph is an arc rotation distance graph.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER ni

F-TRANSFORMATIONS

3.1 A Generalization of Edge Rotation and Edge Slide Transformations

In this chapter we introduce a new type of transformation on graphs, which is a

generalization of the previously defined edge rotation and edge slide. Let G and H

be two (p, q) graphs, both containing a subgraph isomorphic to a given graph F of

order at least 2. We say that G can be transformed into H by an F-rotation (or

simply, G can be F-rotated into H) if there exist distinct vertices u, v and w of G

and a subgraph F' of G isomorphic to F, such that u g V(F'), (v, w) c V(F'),

uv e E(G), uw «É E(G) and H s G - uv + uw. For example, if F = Kj 3 , then the

graph G of Figure 3.1 can be Kj 3 -rotated into H and H'.

y G: H; H':

X X

u z u z

Figure 3.1

More generally, we say that a graph G can be F-transformed into H if either (1) G = H or (2) there exists a sequence G = Gg, G j,..., G„ = H of graphs such

that, for i = 0, 1,..., n - 1, the graph Gj can be F-rotated into Gj+j. For instance,

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36

the graph G of Figure 3,2 cannot h t Kj 4 -rotatedinto H, but G can be Kj^-

transformed into H.

G(= Gq ); G j : H (sG 2 ):

Figure 3.2

Observe that ÏC2 -rotation and K 2 -rotation are edge rotation and edge slide,

respectively. Clearly, if a graph G can be F-transformed into a graph H, then G

and H have the same order, same size, and both contain a subgraph isomorphic to F.

Unfortunately, the converse is not true, in general. For instance, the graphs G and H

of Figure 3.3 have the same order and same size, and both G and H contain a

subgraph isomorphic to C 4 , but G cannot be C 4 -transformed into H. In fact, G

can be C 4 -transformed only into itself.

G: H;

Figure 3.3

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37

3.2 Properties of F-Transformations

One may ask the question: What are necessary and sufficient conditions for one

of two graphs G and H to be F-transformed into the other? We have already seen

the answer to this question if F s ÏC2 or F = K 2 . In this section we study properties

of F-transformations and answer the above question for some connected graphs F.

First we show that if F is a connected graph, then F-transformation preserves

connectedness.

Proposition 3.1 Let F be any nontrivial connected graph. If a connected graph

G can be F-transformed into a graph H, then H is connected.

Proof It suffices to show that if H is obtained from G by an F-rotation, then H

is connected. Without loss of generality, assume that G and H have the same vertex

set and that H = G - uv 4- uw (where, necessarily, v and w belong to a subgraph F'

of G isomorphic to F). Let x and y be two vertices of H. We show that x and y

are connected in H. Since x and y are also the vertices of the connected graph G,

there exists a path P: x =X q , X j,..., x^j = y in G connecting these two vertices. If

an edge uv does not belong to P, then P is also a path in H.

Suppose then that uv is an edge of P. Then, for some 1 < i < n, either

(1) u = Xj_i and v = Xj, or (2) u = Xj and v = Xj_j. Without loss of generality,

assume that u = Xj_^ and v = xj. On the other hand, the vertices u and v belong to

a connected subgraph F' of G; thus Xj_j and must be connected in G by a path,

say P'. Now we can replace an edge xj_jXj of P by the path P', producing a walk

W from X to y. Note that W does not contain edge uv, so it is also a walk in H.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38

Since every walk contains a path having the same end-vertices, the proof is complete. □ An immediate consequence of Proposition 3.1 is the following corollary.

Corollary 3.1a Let F be a nontiivial connected graph. A graph G can be F-

transformed into a graph H if and only if the graph G has components Gj,G2 , ...,

Gjç, the graph H has components Hj, H 2 , ..., and Gj can be F-transformed

into Hj for every i ( 1

Every F-rotation can be viewed as a special edge rotation. Therefore, if a

graph G can be F-transformed into H, then G can be K 2 -transformed into H. If,

additionally, the graph F is connected, then G can also be K 2 -transformed into H

(since if G can be F-transformed into H, then the corresponding components of G

and H have the same order and same size, and therefore, G can be K 2 -transformed

into H). Since ÏC 2 and K 2 are subgraphs of F, the following question can be asked:

If a graph G can be F-transformed into a graph H and F' is a subgraph of F, can

G also be F'-transformed into H? The answer is no. For example, for the graphs F,

F', G, and H of Figure 3.4, H can be obtained from G by an F-transformation, but

not by an F'-transformation. However, the implication holds for graphs F' that are

"uniformly embedded" in F. We say that F' is uniformly embedded in F if, for

every two vertices x and y of F, there exists a subgraph F" of F isomorphic to F'

and containing x and y. For such a subgraph F' of F, every F-rotation is, in fact,

an F'-rotation. As a consequence, we have the following results.

Proposition 3.2 If G can be Kj ^-transformed into H (n ^ 1), then G can also

be Kj ^-transformed into H, for 1 < m < n.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39

Figure 3.4

Next we study properties of Kj ^-transformations.

Proposition 3.3 If G is a tree of order p with A(G) = m, then G can be

Kj ^-transformed into Kj p_j.

Proof Suppose, to the contrary, that G cannot be Kj ^-transformed into Kj p_j.

Among all graphs that can be obtained from G by a Kj ^-transformation, let H be

one whose maximum degree is as large as possible. Let A(H) = d, where, necessarily,

of H adjacent to Vq. Since H is connected and d < p - 1, there exists a vertex u in

H that is adjacent to Vj, for some 1 < i < d, and that is not adjacent to Vq. Define a

new graph H '= H - uv; 4-uvg. Clearly H' is obtained from H by a Kj^-rotation.

Thus G can be Kj ^-transformed into H'. This contradicts our choice of H, since

A(H')>A(H). □

By the transitivity of F-transformations and by Propositions 3.2 and 3.3, we

obtain a more general result.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40

Corollary 3.3a If G and H are any two trees of order p with A(G) < A(H),

then, for 1 < m < A(G), G can be Kj ^-transformed into H.

It turns out that the result of Proposition 3.3 can be generalized to the connected

(p, q) graphs G and H with A(G) = A(H) = p - 1.

Proposition 3.4 Let G and H be two connected (p, q) graphs with A(G) =

A(H) = p - 1, and let m be an integer with 1 < m < p - 1. Then G can be

Kj ^-transformed into H.

Proof If G and H are trees, the result follows from Corollary 3.3a. Assume,

then, that G and H are not trees. Suppose, to the contrary, that G cannot be Kj ^-transformed into H. Among all graphs that can be obtained from G by a

Kj ^-transformation, let H' be one having a maximum number of edges in common

with H. Without loss of generality, we may assume that H and H' have the same

vertex set, say V(H) = V(H') = {vj, V2 >..., Vp).

Since H IT, there exist edges e = VjVj and e' = v^v^ such that e e E(H) -

E(H') and e' e E(H') - E(H). If e and e' are adjacent, define H" = H' - e' + e.

Note that H" is obtained from H' (and therefore from G) by a ^^-transformation

and H has more edges in common with H" than it does with H'. This contradicts

the choice of H'. Suppose then that e and e' are not adjacent. We consider two

cases.

Case 1. Assume that at least one of the four edges v^vj, v^vj, VjVj, and v^vj is not an

edge o f H'. Without loss of generality, we may assume that v^vj does not belong to

H'. Define two new graphs H" = H' - VgV^ + v^vj and H"' = H" - VjVg + VjVj.

Observe that H' is Kj ^-transformed into H", and H" is ^-transformed into

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41

H"'. By transitivity, G can also be Kj ^^-transformed into H'". But H"' has more

edges in common with H than does H', so we again produce a contradiction.

Case 2. Assume that all edges v^vj, v^Vj, v^Vj and v^Vj belong to H'. Then we

define H" = H' - VjVg + VjVj, and H'" = H" - v^v^ + v^Vj. Since H'" is a graph that

can be obtained from G by a Kj ^-transformation and H has more edges in common

with H"' than with H', we have a contradiction again. □

For a vertex v in a graph G we write N(v) to mean the neighborhood of v (the set of vertices adjacent to v) and N[v] = N(v) u (v) for the closed

neighborhood.

Proposition 3.5 Let G and H be two connected (p, q) graphs with m = A(G)

Proof If G and H are trees, the result follows from Corollary 3.3a. Suppose then that G and H are not trees. We show that G (and H) can be ^-transformed

into some graph M with A(M) = p - 1. Then, by Proposition 3.4 and the symmetry

and transitivity of F-transformations, G can be ^^-transformed into H. Suppose,

to the contrary, that G cannot be Kj ^^-transformed to a graph containing a vertex of

degree p - 1. Then, among all the graphs that can be obtained from G by a

K j ^-transformation, let M' be one having the greatest maximum degree and let

A(M') = d.

Let Vq be a vertex of M' with deg^j, Vq = d and suppose Nj^/ (vq) =

(vj, V2 , ..., Vj). Since M' is connected and d < p - 1, there exists a vertex u in M'

that is adjacent to a vertex Vj for some 1 ^ i < d and that is not adjacent to Vq. Define

a new graph M" = M' - uvj + uvq. The graph M" is obtained from M ' by a

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42

K j ^-rotation and therefore G can be Kj^-transformed into M". Moreover,

A(M") > A(M'), which contradicts our choice of M'. □

Now we introduce a transformation on multigraphs, which will be useful in

proving some of the theorems. Let G and H be two (p, q) multigraphs, both

containing a subgraph isomorphic to a given graph F of order at least 2. We say that

G can be transformed into H by a free F-rotation if there exist distinct vertices u, v

and w of G and a subgraph F' of G isomorphic to F such that u g V(F'),

{v, w) c V(F'), uv e E(G), and H = G - uv + uw. More generally, we say that a

multigraph G canhc freely F-transformed mxo H if either (1) G = H or (2) there

exists a sequence G = Gq , G j, ..., Gj, = H of multigraphs such that, for i = 0, 1,...,

n - 1, the multigraph Gj^j can be obtained from G, by a free F-rotation. It is

known (see [ 8 ]) that a graph G can be F-transformed into a graph H if and only if

G can be freely F-transformed into H. For example, the graph G of Figure 3.5 can

be K 2 -transformed and freely K 2 -transformed into H (by the sequence Gq = G,

G j, G 2 = H and Gq = G, H j, H 2 = H of graphs and multigraphs, respectively).

X y u V X y u V

H G 1

X y u V y u VX

Figure 3.5

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43

Our attention now shifts to P^-transformations (where P„ denotes a path on n

vertices). We begin with the following result.

Proposition 3.6 Let G and H be two connnected (p, q) graphs and let m and n be integers with 2 < m < n. If G can be P^-transformed into H, then G can also

be P^-transformed into H.

Proof It suffices to consider the case when H can be obtained from G by a Pjj-rotation. Suppose then, that in G there exists a path P = P„ and distinct vertices

u, V and w such that v and w are vertices of P, the vertex u is not on P, the edge uv is in G, uw is not in G, and H = G - uv + uw. If the distance between vertices

V and w in P is less than m, the P„-rotation transforming G into a graph

isomorphic to H is, in fact, a P^-rotation. Assume then that m < dp(v, w) = k < n.

Let P': V = Xq, Xj, ..., Xj^ = w be a subpath of P and let s = - . For 0 < i < s,

define the graphs (possibly multigraphs for 0 < i < s) Gq, jG, ..., Gg as follows:

Gq = G,

Gi = Gj_i - ux(i_j)(„_j) + ux^n_j), 0 < i < s

and Gg = Gg_i - uXg(^_^) + ux^.

Observe that, for 1 < i < s, Gj is obtained from G j.j by a P^-rotation and Gg = H.

Therefore H can be obtained from G by a free P^-transformation. The desired result

follows. □

Proposition 3.7 Let Tj and T 2 be trees of order p with diamTj < diamT 2 and

let n = diamTj+ 1. Then T^ can be P^-transformed into T 2 .

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44

Proof We show that the tree Tj (and so T 2 as well) can be P„-transformed to the

path Pp. Then, by the symmetry and transitivity of P^-transformations, Tj can be

P^-transformed into T 2 . Suppose, to the contrary, that Pp cannot be obtained from

Tj by a P^-transformation. Among the trees into which Tj can be P^-transformed,

let Tg be one with greatest diameter, and let m = diamTj + 1. Clearly n < m < p.

Also let X and y be two vertices of Tg with d(x, y) = diamTg. Denote the path

connecting x and y in Tg by x = Vq, V j,..., v„, = y. Since Tg is connected, there

exists in Tg a vertex u distinct from Vj, for all 0 < i < m, such that uVj e E(Tg) for

some i with 0 < i < m. Define T 4 = Tg - uvj + uv^. Clearly, Tg is P^-transformed

into T 4 . Hence, by Proposition 3.6, the tree T 4 can be obtained from Tg (and so

from Tj) by a P^j-transformation, which is contrary to the choice of Tg (since

diamTg < diamT 4 >. □

It is the case that not only every tree of order p can be obtained from another such tree by a P^-transformation, but, in fact, every hamiltonian (p, q) graph can be

obtained from another hamiltonian (p, q) graph by a P^-transformation, where

n < p - 1 .

Proposition 3.8 Let G and H be two nonisomorphic hamiltonian (p, q) graphs. Then G can be Pp_|-transformed into H.

Proof Without loss of generality, we may assume that C: Vj, V 2 , ..., Vp, Vj is a

hamiltonian cycle in both G and H. Since G and H are nonisomorphic, there exist chords e = VjVj and e' = VgV^ such that e e E(G) - E(H) and e' e E(H) - E(G). It

suffices to show that the graph G' = G - e + e' can be obtained from G by a

Pp_j-transformation. We consider three cases.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45

Case 1. Assume that the chords e and e' are adjacent. Without loss of generality, let Vj = Vg. Since the graph G - Vj contains a hamiltonian path, the graph G' = G -

V|Vj + VjV^ = G - e + e' is obtained from G by a Pp_j-transformation.

Case 2. Assume that the chords e and e' are not adjacent, and at least one of the

four edges VjVg, vjVj, VjVg, VjVj does not belong to G. Suppose that VjVg is not an

edge of G. Then the graph G" = G - VjVj + vjVg is obtained from G by a

Pp_j-transformation. Furthermore, G" can be Pp_^-transformed into G' = G - e + e'

since G' = G" - + VgV^. Thus G can be Pp_i-transformed into G'.

Case 3. Assume that the chords e and e' are not adjacent and G contains the edges

VjVg, VjVp VjVg, VjVj. Then, the graph G" = G - VgVj + VgV^ can be obtained from G

by a Pp_j-transformation. Moreover, G" can be Pp_j-transformed into G', since

G '= G" - VjVj + VjVg. Hence G can be Pp_j-transformed into G'. □

Proposition 3.9 Let G and H be two connected (p, q) graphs, each containing a path on d (< p) vertices. Then G can be P^-transformed into H.

Proof This result was established in Proposition 3.7 in the case where G and H

are trees. Assume then that G and H are not trees. First we show that G (and so H) can be Pj-transformed into some (p, q) graph containing a hamiltonian path.

Suppose, to the contrary, that G cannot be P^-transformed into a graph having a

hamiltonian path. Then, among all those graphs that can be obtained from G by a

Pjj-transformation, let G' be one containing a longest path and let P; Vj, V 2 , ..., Vj,

be a longest path of G'. Clearly d < n < p. Applying a similar argument as in the

proof of Proposition 3.7, we may conclude that G' can be Pj-transformed into a

graph containing a longer path than any path of G'. This contradicts the choice of G'.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46

Thus there exists a (p, q) graph G' that can be obtained from G by a

Pj-transformation and contains a hamiltonian path, say P: v^, V 2 ,..., Vp. If G' is not

hamiltonian, then G' can be P^-transformed into a hamiltonian graph, as we next

show. We consider two cases.

Case 1. Suppose that degg, Vj > 2 or degg, Vp ^ 2. Without loss of generality,

assume that degg, Vp ^ 2. Then Vp is adjacent to Vj, for some i, 1 < i < p. Observe

that a hamiltonian graph H' = H - VpVj + VpVj is obtained from H by a

Pp_2-transformation; therefore, H' can be obtaind from H by a P^-transformation.

Case 2. Suppose that degQ^ = degQ/Vp = 1. Let i = max{j 1 deg Vj > 3 and 1<

Ho = G',

Hi=Ho-VpVp_i+VpVi, and

^ j +1 " H j - Vp_jVp_j_j + Vp_jVp_j^i, 1 < j < p - i.

Note that, for j = 0, 1,..., p - i - 1, the graph Hj^j is obtained from Hj by a

Pp_j-transformation, and so Hj^j can be obtained from Hj by a P^-transformation.

Moreover, Hp_j contains a hamiltonian path Vj^j, V;^ 2 » •••> Vp, Vj, V2 , ..., Vj with

degc' Vj > 2. Therefore, by the argument from Case 1, the graph Hp_j can be

Pjj-transformed into a hamiltonian graph.

Thus G (and so H) can be P^-transformed into a hamiltonian graph and the

desired result follows (by Propositions 3.6 and 3.8). □

Proposition 3.10 Let F be a connected graph of order p' with 0(F) = 1.

Furthermore, let u be a vertex of degree 1 and v a vertex adjacent to u. If G and

H are two (nonisomorphic) connected (p, q) graphs, each containing F as an

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47

induced subgraph and such that the p - p' vertices not in F are adjacent to v, then G

can be F-transformed into H.

Proof If p = p', the result is obvious. Assume then that p > p' and suppose, to the

contrary, that G cannot be F-transformed into H. Then, among all those graphs that

can be obtained from G by an F-transformation and contain an induced subgraph

isomorphic to F, let G' be one having a maximum number of edges in common with

H. Without loss of generality, we may assume that H and G' have the same vertex

set, namely V(H) = V(G') = {vj, V2 , ..., v^,, Vp/^j, ..., Vp). We may also assume

that F' = <{vj, vg, Vp,))y = ({vj, Vg, ..., Vp,))^, = F with vertices Vj and Vg

corresponding to the vertices u and v of F, respectively. Since H and G' are not

isomorphic, there exist edges e = VjVj and e' = v^v^ such that e e E(H) - E(G') and

e' e E(G') - E(H). Observe that each of the edges e and e' has at most one

end-vertex in F' (since F' is an induced subgraph of G'). Without loss of generality,

let Vj «Ê V(F') and v^ g V(F').

We consider three cases. In each case we produce a contradiction by

constructing a graph H' that can be obtained from G' (and therefore from G) by an

F-transformation and has more edges in common with H than the graph G' has.

Note that the graph Fj = F' - VjV2 + V2 Vj (p' < j < p) is isomorphic to F.

Case 1. Suppose that both edges e and e'have an end-vertex in the set W(F'),that

is, {Vj, vJ Ç V(F'). If vj = Vg, take H' = G' - VjVj + v,v^ = G - VjV^ + v^Vp

Otherwise define graphs

Go = G',

Gi = Go-VjVj + ViVs (where Fg = F),

G 2 = Gj - VgVj + VgV^ (where in this case Fj = F), and

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48

H' = G^. Thus H' can be obtained from G' by a free F-transformation.

Case 2 . Suppose üiat exactly one of the edges e and e' has an end-vertex in V(F'). Without loss of generality, let vj e V(F') and v^<£ V(F'). If the edges e and e' are

adjacent (we may asume that Vj = Vg), then H' = G' - VjV2 + VjV^ = G' - VjV2 + VgV^ is

obtained from G' by an F^-transformation or, equivalently, by an F-transformation.

If e and e' are not adjacent, define

Go = G% Gi = Go-VjVj+ VjVg (where F^sF),

G2 = Gj - VgYj + VgVj (where F. = F),

G 3 = G2 - VgVj + VgV^ (where F^ = F), and

H ' = G 3 .

Thus H' can be obtained from G' by a free F-transformation (through the

sequence Gq, G |, G 2 , G 3 ).

Case 3. Suppose that none of the end-vertices of e and e' belong to V(F'). If e

and e' are adjacent (without loss of generality assume Vj = Vg), then define

Gq = G',

G i = Gq - VjVj + VjV2 (with Fj = F),

G 2 = G - VjV2 + Vjv^ = G - VjV2 + VgV^ (where F^ = F), and

H ' = G2 . On the other hand, if e and e' are not adjacent, then defme graphs

Gq = G',

G i = Gq - VjVj + VjV2 (where Fj = F),

G 2 - Gi - v,V2 + V;Vg (where Fg = F),

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49

G j = G2 - VgVj + VgV 2 (where Fj = F),

G4 = G 3 - VgV2 + VgV^ (where F^ = F), and

H ' = G4 .

Thus in all the cases, we have constructed a graph H that can be obtained from

G' (and so from G) by an F-transformation. Since H' has more edges in common

with H than G' does, a contradiction is produced. □

Proposition 3.11 Let F be a connected graph of order p' with 8 (F) = 1, and let

G and H be two (nonisomorphic) connected (p, q) graphs, each containing an

induced subgraph isomorphic to F. Then G can be F-transformed into H.

Proof Let u be a vertex of F with degp u = 1 and let v be the vertex adjacent to

u. We show that G can be F-transformed into a graph G' containing an induced

subgraph F' isomorphic to F and such that the vertices not in F' are adjacent to the

vertex v' of F' corresponding to the vertex v of F. Since the same argument

applies to H, the result follows from Proposition 3.10. Suppose, to the contrary, that G cannot be F-transformed into G'. Among

the graphs that can be obtained from G by an F-transformation, let G" be one

containing an induced subgraph isomorphic to F and such that the vertex

corresponding to the vertex v of F has the largest degree. Without loss of generality,

we may assume that V(G") = {vj, V2 , ..., Vp,, Vp/^j,..., Vp) and F' = ({vj, V 2 , ...,

Vp'})Q« = F with Vj and V2 corresponding to u and v of F, respectively. Note

that at least one of the vertices Vp/^j, Vp/^2 > •••’ ''p is not adjacent to V 2 . We

consider two cases.

Case 1. Assume that G" has a vertex Vj (p' < i < p) such that ^ E(G") and

VjVj 6 E(G") for some 2 < j < p'. Then the graph G"' = G" - VjVj + V;V2 is obtained

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50

from G" (and so from G) by an F-transformation, has F' = F as induced subgraph,

and degj"' ^2 ^ deg^» V2 , which contradicts our choice of G".

Case 2. Assume that G" has two distinct vertices Vj and Vj (p' < i, j < p) such that

VjV2 SÊ E(G"), VjV2 e E(G") and VjVj e E(G"). Observe that the graph F" = F '-

ViV2 + VjV2 is isomorphic to F. Thus the graph G '" = G" - VjVj + VjV2 is obtained

from G" by an F-transformation and contains F' = F as induced subgraph.

Moreover, deg^»/ V 2 > deg q» 2^- This again produces a contradiction with the choice

of G". □

Next we show that if 6 (F) > 1 or F is not an induced subgraph of G or H,

the result does not necessarily hold. For example, for the graph F of Figure 3.6 we

have 6 (F) = 2 > 1. Although the graphs G and H contain F as an induced

subgraph, G cannot be F-transformed into H. In fact, G cannot be F-transformed

into any graph different from G.

Figure 3.6

For the graph F of Figure 3.7 we have 6 (F) = 1, but F is not an induced

subgraph of G, and H cannot be obtained from G by an F-transformation. Indeed,

G can be F-transformed only into itself and graphs G' and G" of Figure 3.7.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51

F; G: H;

G G";

Figure 3.7

3.3 F-Distance

With each F-transformation described in the previous sections, another metric

can be defined. Let F be a graph of order p' > 2 and let 5 be a set of (p, q) graphs

such that for every pair G, H of graphs in S, the graph G can be F-transformed into

H. The F-distance F-d(G, H) between G and H is defined as the minimum

number of F-rotations necessary to transform G into H. For the graphs F, G and

H of Figure 3.8, we have F-d(G, H) = 2.

G: H:

O O

Figure 3.8

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52

If a graph G cannot be F-transformed into a graph H we set F-d(G, H) =

oo. It is obvious that K 2 -d(G, H) ^ F-d(G, H).

Proposition 3.12 Let F be a graph of order p' ( ^ 2). If G and H are two

(p. q) graphs with p > p' + 2, both containing an induced subgraph isomorphic to F,

then (F u K^)-d(G, H) is defined and

(F u Ki)-d(G, H) < min {F-d(G, H), 2Kz-d(G, H)}.

Proof Without loss of generality, assume that G and H have the same vertex set.

First note that G can be K 2 -transformed into H since G and H have the same

order and same size. Next we show that if a graph H can be obtained from G by a

K 2 -rotation, then H can be obtained from G by an (F u Kj)-transformation.

Suppose then that in G there exist distinct vertices u, v, and w such that uv e E(G),

uw g E(G), and H = G - uv + uw. Observe that G contains an induced subgraph

F' isomorphic to F such that uv g E(F') and uw g E(F'). We consider two cases.

Case 1. Assume that u is a vertex of F'. Then the graph H can be obtained from

G by a free (F u Kj)-transformation through the following sequence of graphs: Go = G,

G |=G o-vu + vw (where F 'u ({w}) = F u Kj),

G2 = Gj - wv + wu (where F' u ({v}> = F v Kj), and

H = G 2 .

Case 2. Assume that u is not a vertex o f F'. If either v or w is a vertex of F', the

edge rotation of uv into uw is, in fact, an (F v Kj)-rotation. Suppose then that

neither v nor w belongs to F'. Let x be an arbitrary vertex of F'. Then H can be

obtained from G by a free (F u Kj)-transformation (through the sequence of

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53

(F U Kj)-rotations resulting in graphs Hq = G, H| = Hq - uv + ux, and H 2 = Hj -

ux + uw = H). In both cases, G can be (F v K 2)-transformed into H and the

(F u Kj)-distance between G and H is defined. Moreover, as we saw, each edge

rotation can be expressed as a sequence of at most two (F u Kj)-rotations; thus

(F U Ki)-d(G, H) < 2K2-d(G, H).

Suppose now that H can be obtained from G by an F-rotation and that H =

G - uv + uw, with u Ê V(F') and {v, w} ç V(F') for some induced subgraph F' =

F. Then H can also be obtained from G by an (F u Kj)-rotation with F' u ({x}) =

F u K j , where x is a vertex (distinct from u) that is not in F'. Therefore, if

F-d(G, H) = 1, then (F u Kj)-d(G, H) = 1, which implies that (F u Kj)-d(G, H) <

F-d(G, H) and completes the proof. □

The following examples show that the presented bound cannot be improved in

general. First we introduce new notation that will be useful for us. Let G and H be

two (p, q) graphs having the same vertex set V = {vj, V2 , ..., Vp}, and let H be a

set of cyclic permutations of the subscripts of the vertices in V. We define the

absolute degree dijference degdif(G, H) of G and H by

P degdif(G, H) = min 2 IdegpVj - degpV^/jJ. Tten i=j

Note that if G = H, then degdif(G, H) = 0, and if H can be obtained from G

by an edge rotation, then degdif(G, H) < 2. Thus, if degdif(G, H) = k, then there are

at least k/2 edge rotations required to transform G into H, that is, degdif(G, H) k

k/2 .

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54

Example 1 Let F be an n-cycle. We define G as the graph obtained firom the cycle

C: Uj, U2 , u „ , Uj by adding n new vertices Wj, W 2 ,W j, and joining Wj and

Uj, for i = 2, 3 , n. Then let H be the graph obtained by identifying some vertex

of C with the central vertex of a star of order n + 1. For n = 5, the graphs G and H

are shown in Figure 3.9. Since degdif(G, H) = 2(n - I), we have F-d(G, H) >

K 2 -d(G, H) > n - 1. Moreover, G can be transformed into H by a sequence of n - 1 edge rotations, namely the rotation of the edge WjUj into WjUj, for i = 2, 3, ..., n;

thus K2 'd(G, H) = n - 1. However, each of these edge rotations is, in fact, a C^-

rotation and a (C^ u Kj)-rotation. Therefore C^-d(G, H) = (C^ u Kj)-d(G, H) =

n - 1. Hence

(F V Ki)-d(G, H) = min {F-d(G, H), 2 K 2 -d(G, H)}.

H; w,

Figure 3.9

Example 2 Let F be a complete graph of order n + 2 (n > 2). Define two graphs

G = K^ ^ 2 ^ " ^ 2 and H = K ^ ^ 2 ^i,n 1 ) ^ 1 . In Figure 3.10 the graphs G

and H are presented for n = 4. Observe that G and H have the same order and

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55

same size, and both contain a subgraph isomorphic to F; however G cannot be F-

transformed into H, so F-d(G, H) = Moreover, K 2 -d(G, H) ^ n - 1 since

degdif(G, H) = 2(n - 1). We show that, in fact, K 2 -d(G, H) = n - 1. Assume that in

G the subgraph isomorphic to nK 2 is induced by the edges UjVj, 1 < i < n. Define

Hq = G and, for i = 1, 2,..., n, define Hj = - U;V| + UjV„. Clearly, the graph

Hj_j can be K 2 -rotated into Hj (1 < i < n) and = H; thus, K 2 -d(G, H) = n - 1.

Note that, for i = 1, 2, ..., n, the graph Hj can be obtained from Hj_j by two

(K^ ^ 2 K^)-rotations, namely Hj_j can be (K ^ ^ 2 Kj)-rotated into =

UjVj + UjW (where w is a vertex of the subgraph of Hj_j isomorphic to K„^ 2 )

Hj' can be (K ^ + 2 Kj)-rotated into H, since H = Hj' - UjW + UjVj,. Consequently,

(K „ ^ 2 Ki)-d(G, H) < 2 K 2 -d(G, H) = 2(n - 1).

Suppose now that H can be obtained from G by a sequence of (IC„^ 2 ^ ^ i ) "

rotations, resulting in graphs G = H q , H j , Hj^ = H (where k < 2(n - 1)). Note

that all graphs Hj (1 < i< k) have a unique subgraph isomorphic to K^+ 2 - Without

loss of generality, we may assume that the graphs H q , H j , ..., Hj^ have the same

vertex set V and the subgraph isomorphic to Kjj ^ 2 is induced by a vertex set V' ç V.

Since the graph Hj can be obtained from Hj_j by a (Kj ^+2 Kj)-rotation, say Hj =

Hj_j - xy + xz, at least one of the vertices y and z belongs to V'. Thus the degree

sequences of Hj_j and Hj differ (by one) in exactly one vertex from each of the sets

V' and V - V'. On the other hand, the degree sequences of G and H differ in n

vertices, all from the set V — V'. In fact, the degrees of n - 1 vertices differ by 1

and the degrees of one vertex in the two graphs differ by n - 1. Therefore, in order to

(K^ ^ 2 Kj)-transform G into H, at least l(n - 1)+(n - 1)1 = 2(n - 1) rotations

are required, which proves that

(K„ + 2 Kj)-d(G, H) = 2(n - 1) = min (K^+ 2 -d(G, H), 2Kz-d(G, H)).

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56

G: "1 0 ------0 Vl

0 ------0 1 w. " 2 ^ 2

0 ------— 0 V3 " 3

" 4 0 ------— 0 ^4

H; " 2

I w.

u ‘3

Figure 3.10

Next we show that every nonnegative integer is the F-distance of some pair of

graphs.

Proposition 3.13 Let F be a graph of order p (> 2) and let n be a nonnegative

integer. Then there exists a pair Gj, G2 of graphs such that F-d(Gj, G2 ) = n.

Proof Let F be a graph of order p > 2 with V(F) = {vj, V 2 , ..., Vp} and

degpVj < degpV 2 < ... < degpVp, and let n be a nonnegative integer. Also let m =

max {n, p, A). We define G as that graph obtained from F by adding, for i = 1, 2,

..., p, a total of m‘ new vertices, each adjacent only with vj. Let Gj and G 2 denote

the graph obtained from G by adding another n vertices Xj, X 2 , ..., x„, each adjacent

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57

only with Vj and Vp, respectively. If we let dj = degpVj, then the degree sequences

of Gj and G2 are:

s(Gj): 1, 1 , 1 , dj + m + n, d 2 + m^, dg + dp_ j + mP “ \ dp + mP, and

s(G2 ); 1 , 1 , 1 , dj + m, d 2 + m^, dg + m^,dp_ j + mP" \ dp + mP+ n.

Moreover, degdif(Gj, G 2 ) = 2n; thus F-d(Gj, G 2 ) ^ K 2 -d(Gj, G 2 ) ^ n.

On the other hand, F-d(Gj, G 2 ) ^ n since G j can be F-transformed into G 2

by a sequence of F-rotations resulting in graphs Hq = Gj and Hj = Hj _ j - XjVj +

XjVp, for i = 1 , 2 , ..., n,. □

It was shown in [3] that for two (p, q) graphs G and H, K 2 -d(G, H) = 1 if

and only if K 2 -d(G + Kj, H + Kj) = 1. We show that this property holds also for

F-distance.

Proposition 3.14 Let F be a graph of order p'^ 2, and let G and H be (p, q)

graphs, where p > p', containing a subgraph isomorphic to F. Then F-d(G, H) = 1

if and only if F-d(G + Kj, H + Kj) = 1.

Proof Obviously, if F-d(G, H) = 1 then F-d(G + Kj, H + Kj) = 1. Suppose then

that F-d(G + Kj, H + Kj) = 1. Without loss of generality, we may assume that

V(G) = V(H). Since H + Kj can be obtained from G + Kj by an F-rotation, there

must exist distinct vertices u, v and w in G and a subgraph F' of G isomorphic

to F such that uv € E(G + Kj), vw g E(G + Kj), u «É V(F'), (v, w) c V(F'), and

the graph H + Kj = G -t- Kj - uv + uw.

Case 1. Assume that there exists a vertex z in G + K j distinct from u, v and w

such that deg( 3 +j^^z = p. Then H = G-uv + uw and F-d(G,H) = l.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58

Case 2. Assume that Case 1 does not occur. Thus at least one of the vertices u, v,

and w has degree p in G + Kj. Certainly, degg^^^u p and degg+j^^w * p.

Thus, V is the only vertex of G + Kj having degree p. Since H + Kj = G + -

uv + uw, the vertex w is the only vertex of H + Kj having degree p and G + -

uv + uw - w = H. Next observe that in G + Kj the vertex v is the only vertex of

degree p (so v is adjacent to every vertex of G) and w is a vertex of degree p - 1

(that is, w is adjacent to every vertex but u), while in H + Kj (that is, in G + Kj -

uv + uw), the vertex w is the only vertex of degree p and v is a vertex adjacent to

every vertex except u. Hence, G + Kj - uv + uw - w = G + Kj - v = G, which

implies that G = H. But then G + Kj = H + Kj, which contradicts the fact that

F-d(G + K j , H + K j ) = 1. □

Corollary 3.14a Let G and H be two (p, q) graphs with F-d(G, H) = 1 and

let p' be an integer with p' > p + 2. Then there exist 2-connected graphs G' and H'

of order p' such that F-d(G', H') = 1.

3.4 F-Distance Graphs

In this section we define a class of graphs related to F-transformations. Let F

be a graph of order p' (^ 2 ) and let 5 be a set of (p, q) graphs, each containing a

subgraph isomorphic to F. Then the F-distance graph (D^iS) of S is that graph

whose vertex set is S and in which two vertices G and H are adjacent if and only if

F-d(G, H) = 1. For example, if F = K 3 and S is the set of graphs G j, G2 , G 3 , and

G4 shown in Figure 3.11, then 2?p(i) = K 4 - e.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59

ô O a

(Dp ({Gj.G^.Gy G^})

° 4

Figure 3.11

It will be useful for us to extend the definition of F-distance graph (Dp{S) to a

set S of graphs not necessarily having the same order and same size. It was shown in

[3] that the union and cartesian product of K 2 -distance graphs are K 2 -distance graphs.

Also graphs, each block of which is a K2 -distance graph, are K 2 -distance graphs.

These are properties of all F-distance graphs as well.

Proposition 3.15 If G and H are two F-distance graphs, then G u H, G x H,

and every graph obtained by identifying a vertex x of G with a vertex y of H are

F-distance graphs.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60

Proof Let G and H be F-distance graphs of order rtij and m 2 , respectively. By

Corollary 3.14a there exist 2-connected (pj, q^) graphs G^, G 2 , ..., and 2-

connected (P 2 > Q2 ) graphs Hj, H2 , ..., with Pi>P2 such that

îZ?p({Gj, G 2 , ..., Gj^^ )) = G and iDp({Hj, H 2 , ..., H ^ ^ )) = H.

It is straightforward to see that (Dp({G|, G 2 , .., G^^, H 2 , ..., H ^^)) = G u H.

Suppose now that V(G) = {uj, U 2 , ..., u^^), V(H) = {vj, V2 , ..., v^^), the

graph Gj corresponds to the vertex Uj (1 < i < mj) and Hj is the graph

corresponding to the vertex Vj (1 < j < m 2 ). Since the graphs Gj (l

Hj (1 < j < m2 ) are 2-connected, F-d(Gj u Hj, Gg u HJ = 1 if and only if either

F-d(Gj, Gg) = 1 and Hj = H^ or Gj = Gj and F-d(Hj, Hj) = 1, that is, if and only if

(1) vertices Uj and Ug are adjacent in G and Vj = v^, or (2) Uj = Ug and Vj is

adjacent with v^ in H. This is equivalent to the condition that in G x H the vertex

(uj, Vj) is adjacent with (Ug, vJ. Therefore, îDp({Gj u Hj 1 1 < i < n, 1 < j < m]) =

G X H.

Since every graph obtained by identifying a vertex x of G with a vertex y of

H is an induced subgraph of G x H, and an induced subgraph of an F-distance graph

is also F-distance graph, the proof is complete. □

Proposition 3.16 Let F be a connected nontrivial graph. Then every graph is an

F-distance graph.

Proof It suffices to show that every connected graph is an F-distance graph (since

then, by Proposition 3.15, the result holds for every graph). Let then G be an

arbitrary connected (p,q) graph with V(G) = {vj, V 2 , ..., Vp). Let also x and y be

two vertices of F with dp(x, y) = diam F. We construct a new graph H as follows.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61

For every edge e of G, let Fg denote a graph isomorphic to F such that if e = VjVj,

then the vertices of Fg corresponding to the vertices x and y of F are denoted by

Xg J and ygj, respectively. For every vertex v of Fg distinct from Xg j and ygj, we

add a new vertex that is adjacent only to v. Then, for i = 1, 2 , p, we identify all vertices of u{V (Fg)le€ E(G)} that correspond to the same vertex Vj of G, and

label this new vertex Wj.

Now, for i = 1, 2,..., p, we add 2i new vertices that are adjacent only with

Wj. Observe that in a graph H constructed as described above, there exist exactly p

vertices (namely, Wj, W2 , ..., Wp) that are adjacent to two or more end-vertices.

Observe also that dj|(wj, Wj) = dgCvj, Vj) diam F.

Next we construct graphs Hj, H 2 , ..., Hp such that F-d(Hj, Hj) = 1 if

do(Vi, Vj) = 1. For i = 1, 2, ..., p, define Hj as the graph obtained from H by

adding a new vertex z that is adjacent only to Wj. The graphs Hj and Hj (i < j)

differ in the numbers of end-vertices adjacent to Wj and Wj. In fact, in Hj there are

21 + 1 and 2 j end-vertices adjacent to Wj and Wj, respectively, while in Hj the

vertices Wj and wj are adjacent to 2 i and 2 j + 1 end-vertices, respectively.

Therefore, if i < j, then F-d(Hj, Hj) = 1 if and only if Wj and Wj belong to a

common subgraph of Hj that isomorphic to F and Hj = Hj - uwj + uwj. If v, and

Vj are adjacent, that is, if e = VjVj e E(G), then Wj and Wj belong to a common

subgraph of Hj that is isomorphic to F (since Wj = Xg j and Wj = yg j, and

(Xg j, ygj) £ V(Fg)) and F-d(Hj, Hj) = 1. On the other hand, if Vj and Vj are not

adjacent in G, that is, dg(Vj, Vj) > 1, then dy(Wj, Wj) = dg(Vj, Vj) diam F > diam F.

Thus the vertices Wj and Wj do not belong to a common subgraph of Hj that is

isomorphic to F and F-d(Hj, Hj) > 1. As a consequence, iDp({Hj,..., Hp}) = G. □

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER rv

TRANSFORMATIONS OF SUBGRAPHS

4.1 Edge Slide Subgraph Transformations

In previous chapters we discussed topics related to transformations of graphs.

In this chapter we turn our attention to transformations of subgraphs. We begin with a

few concepts introduced in [4]. Let Gj and G 2 be edge-induced subgraphs of the

same size in a graph G. The subgraph G2 can be obtained from Gj by an edge

rotation if there exist distinct vertices u, v and w such that uv e E(Gj), uw 4

E(Gi), and G 2 = G^ - uv + uw. More generally, G j can be r-transformed mXo G2

if Gj = G 2 or G 2 can be obtained from Gj by a sequence of edge rotations. It was

shown in [4] that every edge-induced subgraph of a connected graph G can be r-

transformed into any edge-induced subgraph of G having the same size. The distance

dj.(Gj,G 2 ) between Gj and G 2 is the minimum number of edge rotations required to

r-transform Gj into G 2 . For the graph G of Figure 4.1, the subgraph G 3 can be

obtained from Gj by an edge rotation so that d^(Gj,G 3 ) = 1. On the other hand, G3

cannot be obtained from G 2 by an edge rotation, but G 3 can be obtained from G 2

by an r-transformation and dj.(G 2 , G3 ) = 3.

We now introduce a subgraph transformation based on edge slide. Let Gj and

G 2 be two edge-induced subgraphs of the same size in G. We say that G 2 can be

obtained from Gj by an edge slide if there exist distinct vertices u, v, and w of G

such that uv e E(Gj), uw6 ^ E(Gj), vw e E(G) and G 2 = Gj - uv + uw. For

example, for the graph G of Figure 4.2, the subgraph G 2 can be obtained from Gj

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63

G: Gi:

X y tu

w G,: V/

Figure 4.1

by an edge slide. More generally, we say that Gj can be s-transformed into G 2 if

either (1 ) G^ = G2 or ( 2 ) G2 can be obtained from Gj by a sequence of edge

slides.

w w G; Gi

V U V u

z z

Figure 4.2

As we mentioned earlier in this section, for every pair H, H' of edge-induced

subgraphs of the same size in a connected graph G, the subgraph H can be

r-transformed into H'. Unfortunately, this is not the case for s-transformations. For

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64

example, if G is the graph of Figure 4.3, then H cannot be s-transformed into H'.

In fact, H can be s-transformed only into itself.

u w y

Figure 4.3

Next we investigate necessary and sufficient conditions under which one of

two subgraphs of G can be s-transformed into the other. Let G be a graph

with vertex set V(G) and edge set E(G), and let e = uv be an edge not in E(G). Then, by G + e we denote the graph with vertex set V(G) u {u, v) and edge set

E (G )u (e).

Proposition 4.1 Let e and f be two edges of a graph G and let G' be a

subgraph of G - e - f. Then ({e}) can be s-transformed into ((f)) if and only if

G' + e can be s-transformed into G' + f.

Proof If ((e)) can be s-transformed into ((f)), it is straightforward to show that

G' + e can be s-transformed into G' + f.

Conversely, assume that G' + f can be obtained from G' + e by an

s-transformation. Therefore, there exists a sequence G' + e = Hq, Hj, ..., Hjj = G' + f

of subgraphs of G such that, for i = 1 , 2,..., n, the subgraph Hj can be obtained

from Hj_i by an edge-slide. Let Hj = Hj_i - ej + f,, i = 1, 2,..., n. Consider the set

M = (ej, fj 1 i = 1, 2,..., n). Certainly, e and f belong to M, that is, e = ej and f =

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65

fj, for some 1 < i, j < n. Define D as the digraph with vertex set M in which vertex

X of D is adjacent to vertex y if and only if there exists i (1 < i < n) such that Hj =

Hj_ 2 - X + y. Observe that for a vertex x of D distinct from e and f, we have

id X = od X. Furthermore, od e = id e + 1 and id f = od f + 1. Thus, D contains an eulerian e-f trail and, consequently, D contains an e-f path, say P: e = Xq, x^, ...,

x ^ = f. Since (xj_^, Xj) is an arc of D, for i = 1, 2, ..., m, the subgraph ({xj_|})

can be transformed into ({xj}) by an edge slide, and, therefore, ((e)) can be s-

transformed into ((f)). □

Let e and f be edges of a graph G. A triangular e-f walk of G is a finite,

alternating sequence e = eg, Tj, e^, T 2 , ..., e^_j, Tj^, Cj, = f of edges and triangles

such that ej_j and ej belong to Tj (1 < i < n). A triangular e-f patA is a triangular

e-f walk in which no edges or triangles are repeated. The number n of triangles in the

triangular path is called its length. In the graph G of Figure 4.4 there exists a triangular e-f path (with Tj = ((cj_j, ej)), i = 1, 2, 3), but there is no triangular e-g

path.

e = e.

Figure 4.4

It is straightforward to show that every triangular e-f walk in a graph contains

a triangular e-f path.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66

Observe that for every two edges e and e' of a triangle T, the subgraph

<{e')> can be obtained from ((e)) by an edge slide. Therefore, if in a graph G there

exists a triangular e-f path, then = ((e)) can be s-ti'ansformed into G 2 = ((f)).

Whenever edges e and f belong to a 3-cycle in G, we denote this triangle by

T(e, f) and call it a slide induced triangle. With every edge slide there is associated a

unique triangle T, namely, if G 2 = G^ - e + f, then T = T(e, f). These observations

are useful in proving the following result.

Proposition 4.2 Let G be a connected (p, q) graph, and let q' be an integer

with 1 < q ' < q. For every pair G j, G2 of subgraphs of G having size q', the

subgraph Gj can be s-transformed into G 2 if and only if every two edges e, f of G

are connected by a triangular path.

Proof First, assume that for every two subgraphs of G of size q', each can be

s-transformed into the other. Let e and f be two edges of G. We show that in

G there exists a triangular e-f path. Let H be a subgraph of G - e - f having size

q'-l. Define Gj = H + e and G 2 = H + f. Since G^ and G 2 have size q', the

subgraph G| can be s-transformed into G 2 . Thus, by Proposition 4.1, the subgraph

((e)) can be s-transformed into ((f)), that is, there exists a sequence e = eQ, e^,...,

e„ = f of edges of G such that, for i = 1,2, ..., n, the subgraph ( ( e J ) can be

obtained from ({ej_j)) by an edge slide. Therefore, for i = 1, 2, ..., n, the triangle

Tj = T(ei_j, ej), is a subgraph of G.

We claim that if j i + 1 (1 < i < j < n), then the triangles Tj and Tj are

edge-disjoint, for suppose, to the contrary, that for some i and j (with 1 ^ i < j < n

and j ï&i + 1), the triangles Tj and Tj have a common edge. In such a case, if g and

h are edges of Tj and Tj, respectively, the subgraph ((g)) can be transformed into

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67

((h)) by at most two edge slides. But then ((f)) can be obtained from ((e)) by less

than n edge slides (i - 1 edge slides to transform ((e)) into ((ej_i)), at most two

edge slides to transform ((c;_^)) into ((cj)), and n - j edge slides to obtain ((f))

from ((C j)), that is, a total of n - (j - (i + 1)) < n edge slides). Thus, P: e = Cq, Tj,

Cj, T 2 , ..., e^_j, T^, e^ = f is a triangular e-f path.

For the converse, assume that every pair of edges of G is connected by a

triangular path. Let and G 2 be two subgraphs of G having the same size,

namely, q(Gj) = q(G 2 ) = q' < q. Denote the edges of G^ and G 2 by Cj, 0 2 ,..., Cq,,

and f |, f 2 > ..., fq', respectively. Recall that since every pair of edges of G is

connected by a triangular path, then ((cj)) can be s-transformed into ((fj)) for i =

1,2, ...,n. This implies that G| can be s-transformed into G 2 . □

4.2 Triangular Line Graphs

For a given connected graph G, we define its triangular line graph 7(G) as

that graph with vertex set E(G) such that two vertices e and f of 7(G) are adjacent

if and only if T(e, f) is a triangle of G. For G = K 4 - e, the graph 7(G) is shown in

Figure 4.5.

G: 7(G):

Figure 4.5

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68

It follows from the definition that 7(G) is a spanning subgraph of the line

graph L(G). The next result is perhaps less obvious.

Proposition 4.3 Let G be a connected graph of order p ^ 2. Then 7(G) = L(G)

if and only if G = Kp.

Proof If G = Kp, p > 2, then every two adjacent edges belong to a triangle; thus

L(G) = 7(G). Assume now that 7(G) = L(G). Then, for every pair e, f of adjacent

edges in G, there exists a triangle containing both e and f. Suppose, to the contrary,

that G is not a complete graph. Let A = A(G) and let v be a vertex of maximum

degree. Since every two adjacent edges belong to a triangle, (N[v]) = ^ j. The

graph G is connected but not complete; thus there exists a vertex x 4 N[v] that is

adjacent to some vertex y of N(v). But then deg y > A , which produces a

contradiction. □

The connectedness of 7(G) is a necessary and sufficient condition for a

subgraph of G to be s-transformed into another subgraph of the same size. To show

this, it suffices to prove this fact for edge-induced subgraphs of size 1 .

Proposition 4.4 Let G be a connected nontrivial graph. For every pair Gj, G 2

of edge-induced subgraphs of G having size 1, the subgraph Gj can be s-

transformed into G 2 if and only if 7(G) is connected.

Proof Suppose, for every pair G^, G 2 of edge-induced subgraph of G having size

1, that Gj can be s-transformed into G 2 . Let e and f be two distinct vertices of

7(G) (and therefore distinct edges of G). Define Gj and G 2 as the subgraphs of G

induced by e and f, respectively. Since Gj can be s-transformed into G 2 , there

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69

exists a sequence Gj = Hq, Hj,H,, = G2 (n ïï 1 ) of edge-induced subgraphs of

G such that Hj can be obtained from Hj_| by an edge slide for all i (l< i^ n ).

Suppose that for i = 0, 1, n, the subgraph Hj = <{e;)> for some ej e E(G),

where eg = e and e„ = f. Then, for i = 1, 2 , n, the edges ej_j and ej belong to

a common 3-cycle and therefore ej_j and ej are adjacent vertices of V(G). Thus,

P: e = eg, C j,..., - f is a path of KG) connecting e and f.

For the converse, assume that KG) is connected. Let Gj and G 2 be edge-

induced subgraphs of G having size 1, say G^ = ((e)) and G 2 = ((f)) for some

edges e and f of G. Since 1 (G) is connected and e and f are vertices of 1 (G),

there exists a path P in 1 (G) connecting e and f, say P: e = Cg, Cj, ..., Cj, = f.

Thus, ej_j and Cj (1 < i < n) belong to a common 3-cycle in G and, therefore, the

subgraph Hj_j = ({ej_|}) of G can be transformed into the subgraph Hj = ({ej)) by

an edge slide. Consequently, G | can be s-transformed into G 2 . □

As an immediate consequence of this result, we have the following.

Corollary 4.4a Let G be a connected nontrivial graph. Then for every pair G|,

G2 of edge-induced subgraphs of G having the same size, G ^ can be s-transformed

into G 2 if and only if 1 (G) is connected.

The previous result can be generalized further.

Corollary 4.4b Let G be a connected graph of size q, and let 1 (G) have

k components Tj, T 2 , ..., T,^. Then for two edge-induced subgraphs G^ and G 2

of G having the same size, Gj can be s-transformed into G 2 if and only if

I E(Gi) n V(T;) I = I E(G 2 ) n V(T;) I for i = 1, 2,..., k.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70

Prior to stating the next result, we define a cycle basis of a (p, q) graph G. Let

S(G) denote the vector space of edge-disjoint unions of cycles of G (together with the

empty set 0 ) over the field Z 2 . The addition 0 is defined on S(G) as follows: if

C j, C2 6 S(G), then Cj 0 C2 is the cycle (possibly edge-disjoint union of cycles)

induced by the symmetric difference of the edge sets E(Cj) and E(C 2 ). The

dimension of S(G) is denoted by dim S(G). It is well known that if G is connected,

then dim S(G) = q - p + 1. A cycle basis ®(G) of G is defined as a basis for the

cycle space S(G) in which every element is a cycle. Atriangular cycle basis is a cycle

basis in which every cycle is a triangle. For example, the graph G of Figure 4.6 has

four cycles Cj: Uj, U2 , u^, Uj: C 2 : U2 , Ug, u^, U 2 ; C 3 : Uj, U2 , U3 , U4 , Uj; and C 4 : u^,

U5 , Ug, U4 . The set (3j(G) = {Cj, C3 , €4 } is a cycle basis of G and (82(0 ) = {Cj,

C2 , C4 } is a triangular cycle basis.

Figure 4.6

Certainly, every element C of S(G) (in particular, every cycle of G) can be

uniquely expressed as a linear combination of cycles from a given cycle basis (3(G) =

{Cj, C2 , ..., C„}, namely, C = ttjC j 0 tt 2 C 2 0 ... 0 cx„C„, where ttj e (0, 1) for

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71

0 i < n. For example, if G is the graph of Figure 4.4 and we consider 0^(G) as a

are called the coordinates of C (with respect to ®j(G)).

Proposition 4.5 If G is a 2-connected graph having a triangular cycle basis, then

*2(G) is connected.

Proof Let G be a 2-connected graph with triangular cycle basis ® = {Tj, T 2 , ...,

T„}, and let e and f be distinct vertices of 1 (G) (and so are distinct edges of G).

Since G is 2-connected, there exists a cycle C in G containing both edges e and f. In order to show that there exists an e-f path in 7(G), we employ induction on the

number k of positive coordinates of C. If k = 1, then C is a triangle and so e and f are adjacent vertices of 7(G).

Assume that if e and f belong to a cycle with at most k positive coordinates, then

there exists a path in 7(G) connecting vertices e and f. Suppose now that e and f

belong to a cycle C whose k + 1 coordinates are positive. Without loss of generality,

E(Tk^l). Observe that the triangle has at least one edge, say e', that does not

Then, by the inductive hypothesis, there exists a path in 7(G) connecting e and e'.

On the other hand, e' and f are adjacent vertices of 7(G); thus, there exists a path

connecting e and f. □

The converse of Proposition 4.5 does not hold. For example, if G is the

graph shown in Figure 4.7, then 7(G) is connected; however, there is no triangular

cycle basis for G.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72

G: 7(G):

Figure 4.7

Complete graphs (n > 3), wheels (n ^ 3), and maximal outerplanar

graphs are examples of graphs with triangular cycle bases. We next describe another

family of graphs having triangular cycle bases. A graph G is called a chordal graph if

every induced cycle of G is a triangle. If G is a chordal graph, then for every m-

Proposition 4.6 Every chordal graph with cycles has a triangular cycle basis.

Proof Let G be a chordal graph with at least one cycle. Suppose, to the contrary,

that G does not have a triangular cycle basis. Let tB(G) = {Cj, C 2 , ..., C„) be a

cycle basis of G having a maximum number of triangles. Without loss of generality,

assume that ®i(G) = {Cj, C2 , ..., Cj^), k < n, is the set of all 3-cycles of ®(G). Let

bean m-cycle (m > 3). Since G is a chordal graph, there exist triangles Tj,

element of a cycle basis, thus, at least one of the tiiangles Tj, T 2 , ..., T^ _ 2 cannot be

expressed as a linear combination of cycles from 3^(G). Let Tjp be such a triangle.

Then @2 (G) = ®i(G) u {Ti^} is a set of independent cycles of G and, as such, can

be extended to some cycle basis ^B'(G). This contradicts the choice of (8(0) since

(G) has more triangles than (8(0) does. □

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73

Corollary 4.6a Let G be a 2-connected chordal graph. Then for every pair G,,

G 2 of edge-induced subgraphs of G having the same size, Gj can be s-transformed

into G 2 .

Proof This follows immediately from Propositions 4.5 and 4.6 and Corollary 4.4a. □

Observe that the converse of Proposition 4.6 does not hold. For example, the

graph G of Figure 4.8 has a triangular cycle basis @(G) = {Cj, C 2 , C3 , C4 , C5 , Cg)

(where C^: v^, V2 , Vg, Vp C 2 : v^, V2 , Vg, v^; Cg: Vg, v^, Vg, Vg; C 4 : v^, Vg, v^, v^;

C 5 : Vj, vg, V 6 , Vj; and C^: Vj, Vg, v^, Vj), but is not chordal (since G contains the

induced 4-cycle V 2 , Vg, v^, Vg, V 2 ).

Figure 4.8

To describe necessary and sufficient conditions for the connectedness of

7 (G), we present the cycle basis graph (introduced as the cycle graph with respect

to the cycle basis (B by Syslo [12]). Let G be a 2-connected graph with cycle basis 3(G). The cycle basis graph CBg(G) of G with respect to 3(G) is that graph

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74

whose vertex set is (B(G), and in which two vertices are adjacent if and only if the

corresponding cycles have a common edge. For the graph G s W j 4 of Figure 4.9, if

we let Cj: Vq, Vj, V2 , Vq; C2 : Vg, Vg, Vg, Vg; C 3 : Vq, V3 , V4 , Vq; C4 : Vq, Vj, V4 , Vq;

Cg: VQ, vj, V 4 , V3 , Vq; and Cg: Vq, Vj, Vg, V 3 , Vq, then for = {Cj, C 2 , C3 , C4 },

the cycle basis graph is CB(g^(G) = C4 , while for @ 2 = C3 , Cg, Cg), we have

CBr^iG ) = K4 — e .

Figure 4.9

Proposition 4.7 Let G be a graph every edge of which belongs to a triangle,

and let (3(G) be a cycle basis of G with a maximum number of triangles. Then every

edge e of G belongs to some triangle of 0(G).

Proof Let e be an edge of G, and let be a triangle of G containing e.

Suppose, to the contrary, that ^ 0(G). Since 0(G) is a cycle basis with a

maximum number of triangles, can be expressed as the linear combination of

elements of 0^(G), where 0^(G) denotes the set of triangles of 0(G). Suppose that

there exists an integer i, with 1 < i < k, such that e e Cj, which completes the proof.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75

Proposition 4.8 Let G be a graph for which *Z(G) is connected and let (B(G) be

a cycle basis of G, If a 3-cycle T of G can be expressed as a linear combination of

k with V (‘Tj) = U E(Ci) and ECTj) = {ef 1 T(e, f) = Cj for some 1 ^ i < k) is 1 = 1 connected.

Proof Let e and f be two vertices of rTj, that is, there exist integers i and j with

1 < i, j < k such that e e Cj and f e Cj. Without loss of generality, we assume that

e 6 C] and f e Cj.. To show that ‘Tj is connected, we employ induction on the

number k of cycles in the linear combination T = Cj ® C 2 ® ... ® Cj^.

If k = 1, the result is trivial. Suppose then, that T j is connected for every

triangle T that has at most k triangles of (B(G) in its linear combination. Now let T

be a 3-cycle with T = Cj ® C2 ® ... ® C|^^j. If we write T' = Cj ® C2 ® ... ® Cj^,

then T = T'®Cj^^i. Let e' be a common edge of T' and (certainly, such an

edge exists since T 9^ By the inductive hypothesis, 2*^» is connected. Thus

there exists an e-e' path m ‘Ijf (and so in Tj). Furthermore, e' and f belong to

so c' and f are adjacent in ‘Tj. Consequently, there exists a path in

connecting e and f. □

Proposition 4.9 Let G be a 2-connected graph and ®(G) a cycle basis of G

with a maximum number of triangles. Then ‘1(G) is connected if and only if (1) every edge of G belongs to a triangle and (2) the subgraph of CB^G) induced

by the vertices corresponding to triangles is connected.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76

Proof Let G be a graph for which 7(G) is connected and let ®(G) be a cycle

basis with a maximum number of triangles. Denote by ®^(G) the set of all triangles of

®(G). Since 7(G) is connected, every edge of G belongs to a triangle. It remains to

show that the subgraph of CB^(G) induced by the vertices corresponding to the

elements of ®^(G) is connected. Let T and T be distinct vertices of CBg(G) with

T, T' e ®j(G), and let e and f be distinct edges of the triangles T and T',

respectively. Since 7(G) is connected, e and f are connected in G by a triangular

path e = eQ, T j, e^, T2 , ..., e„_j, T„, e„ = f. We consider two cases.

Case 1. Assume that Tj e (B^{G),for i = 1, 2, ..., n. Then P: T j, T2 , ..., is a path in CBg(G). Moreover, T and have the edge e in common, and f is a

common edge of T' and T^. Thus T and T' are either vertices of P or are adjacent to vertices of P. In either case, there exists a path in CBg(G) connecting T and T.

Case 2. Assume that Tj ^ @^(G) for some 1 < i < n. Since (B(G) is a cycle basis

with a maximum number of triangles, Tj can be expressed as a linear combination of

the elements of ‘BfG). Suppose that Tj = C | 0 C2 ® ... 0 Cj^, where for j = 1, 2,

..., k, Cj e (BfG). Recall that Tj = T(ej_j, Cj). Without loss of generality, we

may assume that ej_j e Cj and e■^ e Cj^. By Proposition 4.8 the subgraph 7^. of

7(G) is connected. Therefore, there exists a triangular ej_j-ej path P,: ej_| = fg, T ^,

f,. t Ç T®, = e,, where T>, .... T,® e ®^(G). Thus every

triangular subpath ej_j, Tj, ej of P for which Tj ^ (BfG) can be replaced by the

triangular path Pj producing a triangular e-f walk W. Since every triangle in W

belongs to (BfG) and every triangular e-f walk contains a triangular e-f path, there

exists a triangular e-f path in which every triangle belongs to (B{G). Denote this path

by F : e = f^, T{, f{, T^, ..., fgj,, t;, f^ = f. Then P": T{, T^, ..., T' is a path in

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77

CBg(G). Similarly as in Case 1, T and T' are either vertices of P" or are adjacent to

some vertices of P", implying that there exists in CB^G) a path connecting T and

T'.

Conversely, suppose that conditions (1) and (2) are satisfied. Let e and f

be two distinct vertices of KG). By Proposition 4.7, there exist triangles T and T'

in 3(G) such that e e E(T) and f e E(T'). Since CBg(G) is connected, the

vertices T and T' of CB^G) are connected by a path. Let P: T = T q , T j ,..., Tj^ =

T' be a path connecting T and T'. Then, for i = 1, 2,..., k, the triangles Tj_j and

Tj have a common edge, say e,. Thus e, T q , e^, Tj, e2 , ..., ej^, Tj^, f is a triangular

e-f walk in KG). Since every triangular e-f walk contains a triangular e-f path, the

desired result follows. □

For integers n > 2, the nth iterated triangular line graph T"(G) of a graph G

is defined to be *3(‘T"“^(G)), where T^G) denotes 7(G) and T"~^(G) is assumed

to be nonempty. Clearly, T"(G) is a subgraph of the nth iterated line graph L"(G)

of G. In fact, for n = 1, T ^G ) = KG) is a spanning subgraph of L^(G) = L(G).

Note that every triangle T in G gives rise to a triangle T ' in 7(G) with a

one-to-one correspondence between the edges of T and the vertices of T'. Moreover,

if Tj and T 2 are two triangles of G, then the corresponding triangles Tj and T 2 of

KG) are edge-disjoint. For suppose, to the contrary, that Tj and T 2 have an edge in

common or, equivalently, Tj and T 2 have two common vertices, say e and f.

Necessarily, e and f are common edges of Tj and T 2 , which implies that Tj =T2 .

Thus 7(G) has at least as many triangles as G has. We show that only for K^-free

graphs G are the number of triangles in G and 7(G) equal.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78

Proposition 4.10 Let t(G) denote the number of triangles in a graph G. Then t(G) = tCZ(G)) if and only if G is K^-free.

Proof Let G be a graph with t(G) = t('7(G)). Suppose, to the contrary, that G

contains a subgraph H = K^, and let V(H) = {Vj, V 2 , V3 , V4 } and E(H) = {ejj =

VjVj 11 ^ i < j < 4}. Then, for 1 < i < j < 3, the vertices e^^ and ej 4 are adjacent in

T(G) (since the edges 6 5 4 , ej4 , and e,j induce a triangle in G) and

<{ei4 , e2 4 , 6 3 4 )) = K3 . However, there is no corresponding triangle in G, as the

edges ej 4 , e2 4 , and eg 4 induce the subgraph isomorphic to 3 instead of a triangle.

For the converse, it suffices to show that t(7(G)) < t(G). Let G be a K 4 -free

graph, and let T be a triangle of T(G) with V(T) = {e, f, g). Then e, f, and g are

pairwise adjacent edges of G, that is, the three edges induce a subgraph isomorphic

either to K3 or Kj 3 . Moreover, every two of the three edges e, f, and g belong to

a common triangle. Thus the end-vertices of e, f, and g induce either K 3 or K 4 in

G. Since G is K 4 -free, the edges e, f, and g induce a triangle in G, implying that

Proposition 4.8 is useful in showing that, for a K 4 -ffee graph G and for n k

3, the nth iterated triangular line graphs of G are isomorphic.

Proposition 4.11 Let G be a K4 -free graph. Then T"(G) = T^(G), for n ^ 2.

Proof Assume that G is a K4 -free graph. Then every edge of ‘ 2(G) belongs to

exactly one triangle. This implies that “IflfG)) has t(‘2 (G)) components, each of

which is isomorphic to K3 . Since ‘2(Gj v G 2 ) = *2 (Gj) v 'KG2 ) and ‘ 2(K 3 ) = K 3 ,

the desired result follows. □

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79

The previous result does not hold for a graph G = K^. However, T (K^) =

8 K 3 and, therefore, for n ^ 3 we have T"(K^) = T^(Kg). The graphs G = and

T'(K^), 1 < 1 < 3, are shown in Figure 4.10. We close this section with the following.

Conjecture For every graph G containing at least one triangle, there exists an

integer k > 0, such that for n ^ k, T"(G) = T^(G).

7(G):

T^(G):

Figure 4.10

4.3 An Introduction to Subgraph Slide Distance Graphs

The n-subgraph distance graphs were introduced in [4]. Let G be a graph of

size q 1) and let n be an integer with 1 < n < q. The n-subgraph distance graph L„(G) of G is that graph whose vertices correspond to the edge-induced subgraphs

of size n in G and where two vertices of L^^(G) are adjacent if and only if the

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80

r-distance between corresponding subgraphs is 1. It is convenient to label the vertices of L„(G) by the edge sets of the corresponding subgraphs or simply by listing the

edges. Each edge in a vertex label is called a coordinate. Since the coordinates are

elements of a set, the order in which the coordinates of a vertex are listed is irrelevant. For example, if a vertex of L^^(G) corresponds to the subgraph of G induced by the

edge set {e^, e 2 , ..., e„}, then we may label this vertex as ej, e2 , ..., e^^ or ej v X,

where X = {ej | 1 < j < n, j i}, or simply as ejX. For the graph G = Kj + (Kj u

K2 ) of Figure 4.11, the graphs Lj(G), i = 1, 2, 3, 4, are shown.

bed

acd abd ac

abc

bd cd

L/G): abed O

Figure 4.11

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81

The graphs L^(G), 1 < n ^ q = E(G), are also called generalized line graphs

since the 1-subgraph distance graph L|(G) is the line graph of G. We shall also refer

to these graphs as n-subgraph rotation distance graphs to distinguish them from the n-

subgraph slide distance graphs, which we are about to describe. We begin with the

definition of n-subgraph slide distance.

Let G be a graph of size q (> 1), and let Gj and G 2 be two edge-induced

subgraphs of G having the same size n (1 < n < q). We define the n-subgraph slide

distance dg(Gj, G 2 ) between Gj and G 2 as the smallest nonnegative integer k for

which there exists a sequence Hg, Hj, ..., of subgraphs of G such that Gj =

Hq, G2 = and, for i = 1 , 2 ,..., k, Hj can be obtained from Hj_j by an edge

slide. If no such k exists, we define dg(Gp G 2 ) = If G = - e, and Gj and

G 2 are two subgraphs of G shown in Figure 4.12, then dg(G|, G 2 ) = 2.

G: G j:

o d

Figure 4.12

We define the n-subgraph slide distance graph S„(G) of G as the graph

whose vertices correspond to the edge-induced subgraphs of size n and where two

vertices Gj and G 2 of Sj^(G) are adjacent if and only if dg(G|, G 2 ) = 1. It is

straightforward to see that Sj(G) = "7(G), and, therefore, Sj(G) is a spanning

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82

subgraph of LjCG). In general, for n 5:1, 8^(0) is a spanning subgraph of L„(G).

For the graph G = - e, the graphs S;(G), 1 < i < 5 are shown in Figure 4.13.

G: h S i(G ):

d c

SgCG): 8 3 (G):

ae abc bee bed

ae ab bde ede abe ce abd ade aed bd ace

8 4 (G): 85(G ) = K i: bode acde abede O abed

abce abde

Figure 4.13

Observe that for the graph G = K4 - e, we have 8 2 (G) = 8 3 (G) and 8 j(G) =

8 4 (G). This fact can be generalized as follows.

Proposition 4.12 Ixt G be a graph of size q (5 1) and let n be an integer with

1 < n < q. Then 8 „(G) = 8 q_„(G).

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83

Proof Define a mapping from the vertex set of S„(G) onto the vertex set of

Sq_n(G) by <|)((e^, e 2 >.... e„}) = E(G) - (e^ 0 2 , e^}. Clearly, (|) is a one-to-one

mapping. We show that

Sjj(G) are adjacent if and only if X = eU and Y = fU, where e, f ^ U, e f, and | u I =k-l (that is, X and Y differ in exactly one coordinate), and e and f belong

to a common 3-cycle in G. Now consider two adjacent vertices X and Y of S^(G).

Then X and Y differ in exactly one coordinate, say X = eU and Y = fU, with e f

and IU I = n - 1. Equivalently, (t)(X) = fU" and (|)(Y) = eU" with lu"l = q-n-l (since <|)(X) = (|)(eU) = E(G) - U - { e ) = U '- { e ) and

{f} = U' - (f); by taking U" = U' - (e, f) we have (j)(X) = U" u (f) = fU" and

(j)(Y) = U" u (e) = eU"). Therefore, X and Y are adjacent in S^(G) if and only if

(])(X) and <})(Y) are adjacent in Sq_„(G). □

4.4 Some Problems Concerning Subgraph Distance Graphs

In [4] and [14] the planarity of subgraph rotation distance graphs was

investigated and the next three results were established. Recall that Lj(G) = L(G); thus

necessary and sufficient conditions for a planar graph G to have a planar line graph are

given by Sedlâ£ek's theorem (see [14]).

Proposition 4A Let G be a planar graph. Then L(G) is planar if and only if

A(G) < 4 and if deggv = 4, then v is a cut-vertex.

This result was extended in [4] by determining all connected graphs for which

L 2 (G) is planar.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84

Proposition 4B Let G be a connected graph. Then 1-2 (0 ) is planar if and only if

either 0 s (n ^ 3) or G is a subgraph of one of the six graphs of Figure 4.14.

^1- O O Go:

Ô 6 o ---- o— o

G 3 : G.: O O

Figure 4.14

This result was extended further in [5] by determining graphs for which L„(G)

(3 S n < q - 3) is planar. By the graph P^, where 1 < i < ^, we mean the tree

obtained by joining a new vertex to a vertex of P„ at distance i from an end-vertex of

Pn-

Proposition 4C Let G be a connected graph of size q and let n be an integer

with 3 < n < ^. Then L„(G) is planar if and only if n = 3 and G is isomorphic to

either P7 or Pg.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85

The problem of determining the planarity of n-subgraph slide distance graphs

appears to be more complicated. Recall that S„(G) is a spanning subgraph of L^(G).

Therefore, if L^(G) is planar, the graph S„(G) is planar as well. It turns out that the

class of graphs G for which L^(G) is planar is a proper subset of the class of graphs

G for which S^(G) is planar. For example, if G s Wj (m > 5) then, for 1 ^ n <

the graph S„(G) is planar while L^(G) is not. We show that G is required to

be Kg-free in order for Sj(G) to be planar.

Proposition 4.13 If S|(G) is planar, then G does not contain a subgraph

isomorphic to Kg.

Proof Suppose, to the contrary, that G has a subgraph H = Kg. Certainly S^(H)

is a subgraph of S^(G). Furthermore, degg^^^^e = 6 for every vertex e of Sj(H)

(since the edge e of H belongs to three triangles, and therefore, the vertex e of

Sj(G) belongs to three edge-disjoint triangles) which implies that Sj(H) (and so

8 2 (G)) is nonplanar. □

Let H be a graph. If for every graph G containing a subgraph isomorphic to

H, the graph 8 2 (G) is nonplanar, we call H a forbidden subgraph. Thus, Kg is a

forbidden subgraph. Since Kg - e is not a forbidden subgraph ( 8 2 (K g -e) is planar

as shown in Figure 4.15), we may call the graph Kg a minimal forbidden subgraph.

The converse of Proposition 4.13 does not hold. For example, the graph G

of Figure 4.16 is Kg-free but 8 2 (G) is nonplanar since it contains a subgraph

homeomorphic from Kg (see Figure 4.16).

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86

K s - e : Si(Kg-e):

Figure 4.15

Figure 4.16

One may expect that if Si(G) is to be planar, then the graph G must be

planar. However, this is not the case. For example, the graph G = 3Kj + P 3 is

nonplanar since it contains a subgraph isomorphic to K 3 3 , but Sj(G) is planar as

illustrated in Figure 4.17.

It is well known that if G is a planar (p, q) graph with p ^ 3, then q < 3p -

6 . Since p(Si(G)) = q(G) and 2q(S^(G)) = ^ degs,(G)e = 2 ^ t(3,e, ceE(G) c6E(G)

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87

it follows that if S^(G) is planar, then ^ tQ.e ^ 3q(G) - 6 . Note that if G c e E ( G )

Kg - e, then (G,e = 3q(G) - 6 . e e E(G)

Figure 4.17

Next we investigate graphs G for which Sj(G) is eulerian or hamiltonian.

Proposition 4.13 Let G be a nonempty graph. Then every nontrivial component

of Sj(G) is eulerian .

Proof Consider a vertex e of Sj(G). It suffices to show that degg^^Q^e = 2n,

for some integer n > 0, If the edge e of G does not belong to a triangle, then

degSj(G)e = 0. Suppose that e belongs to a triangle T, with E(T) = {e, f, g}. Then,

necessarily, the vertex e of Sj(G) is adjacent to the vertices f and g. Thus, if the

edge e belongs to n triangles in G, then the degree of the vertex e in S^(G) is 2n. □

The previous result can be generalized as follows.

Proposition 4.14 Let G be a graph of size q > 1 and let n be an integer with

1 < n ^ q. Then every nontrivial component of S„(G) is eulerian.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88

Proof Suppose that edges e, f, and g induce a triangle in G. Let

Mx(e,f,g)={eX,fX,gX|xcE(G)-{e,f,g} and |x|=n-l),

and

Ny(e, f, g) = {efY, egY, fgY | Y C E(G) - {e, f, g} and | Y | = n - 2).

Every element of a set M^fe, f, g) or N(e, f, g) can be viewed as a vertex of SjfG).

Define Tx(e, f, g) = (Mx(e, f, g)>Sj(G) and Ty(e, f, g) = Sj(G)- We

prove that S^(G) is Kg-decomposable by showing that (1) Tx(e, f, g) = Kg and

T y(e, f, g) = K 3 , and (2) every edge of Si(G) belongs to exactly one of the triangles

of the set A u A, where

A = {Tx(e, f, g) l<{e, f, gDg S K 3 , X G E(G) - (e, f, g}, 1x1 = n - 1), and

A = (Ty(e, f, g) 1 ((e, f, g))^ = K 3 , Y G E(G)- (e, f, g), 1y1 =n-2}.

Since the edges e, f, and g induce a triangle in G, for every set X G E(G) with

1x1 = n - 1, the vertices eX , fX, and gX of Sj(G) are pairwise adjacent.

Similarly, the vertices efY, egY, and fgY are pairwise adjacent. Therefore,

T x(e,f, g) = K 3 and T y(e, f, g) = K 3 .

Now let UW be an edge of Sj(G). Thus, the vertices U and W have n -1

common coordinates, that is, U = eX and W = fX, for some edges e, f of g, and

X g E(G) with 1x1 = n - 1. Moreover, the edges e and f belong to the

cycle T(e, f). Let g be the third edge of T(e, f). If g g! X, then the vertices U, W,

and Z = gX belong to Mx (e, f, g) and the edge UW belongs to the triangle

Tx(e, f, g). On the other hand, if g e X, let Y = X-(g). Since | Y | = n - 2 and

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89

Y C E(G) - {e, f, g), the vertices U, V, and Z belong to Ny(e, f, g) and UW is an

edge of Ty(e, f, g). The triangle to which UW belongs is determined according to

whether g is in X; thus UW cannot be in both T^Ce, f, g) and Ty(e, f, g).

Therefore, the graph S^(G) is Kg-decomposable. The desired result follows. □

Observe that, for n > 3, the graph Sj(K„) is hamiltonian. It follows from the

fact, that Si(K„) = L(K^) and the line graph of a complete graph of order p > 3 is

known to be hamiltonian. However, a characterization of graphs G for which Sj(G)

is hamiltonian remains an open problem.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. REFERENCES

[1] V.Balâl, J. Ko£a, V. Kvasni£ka, and M. Sekanina, A metric for graphs. C&sopis Pësî. Mat. 110 (1986) 431-433.

[2] G. Benadé, W. Goddard, T. A. McKee, and P. A. Winter, On distances between isomorphism classes of graphs. To appear in Casopis Pest. Mat.

[3] G. Chartrand, W. Goddard, M. A. Henning, L. Lesniak, H. Swart, and C. E. Wall, Which graphs are distance graphs? Ars Combinatoria 29A (1990) 225- 232.

[4] G. Chartrand, H. Hevia, E. B. Jarrett, F. Saba, and D. V/. VanderJagt, Subgraph distance and generalized line graphs. To appear in Proceedings of the Second China-USA International Conference in Graph Theory, Combinatorics, Algorithms and Applications San Francisco State University (1991).

[5] G. Chartrand, H. Hevia, E. B. Jarrett, and D. W. VanderJagt, Planarity of n- subgraph distance graphs. Advances in Graph Theory (ed. V.R. Kulli) Vishwa International Publications, Gulbarga, India.

[6 ] G. Chartrand, G. L. Johns, K. S. Novotny, and O. R. Oellermann, Subgraph distance in graphs. To appear in Journal o f Combinatorics, Information & System Sciences.

[7] G. Chartrand, F. Saba, and H. B. Zou, Edge rotations and distance between graphs. Casopis Pèst. Mat. 110 (1985) 87-91.

[8 ] R. J. Faudrcc, R. II. Schelp, L. Lesniak, A. Gyârfâs, and J. Lehel, On the rotation distance of graphs. Submittted to Discrete Math.

[9] M. A. Johnson, Relating metrics, lines and variables defined on graphs to problems in medicinal chemistry. Graph Theory with Applications to Algorithms and Computer Science (eds. Y. Alavi, G. Chartrand, L. Lesniak, D. R. Lick and C. E.Wall) Wiley, New York (1985) 457^70.

[10] M. A. Johnson, An ordering of some metrics defined on the space of graphs. Czech. Math. J. 37 (1987) 75-85.

[11] J. Sedlâbek, Some properties of interchange graphs. Theory of Graphs and Its Applications. Academic Press, New York (1962) 145-150.

[12] M. M. Syslo, On characterizations of cycle graphs. Colloque CNRS, Problèmes Combinatoires et Théorie des Graphes. Orsey (1976), Paris (1978) 395-398.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91

[13] B. Zelinka, On a certain distance between isomorphism classes of graphs. CasopisPêst. Mat. 100 (1975) 371-373.

[14] B. Zelinka, A distance between isomorphism classes of trees. Czech. Math. J. 33 (108) 1983, 126-130.

[15] B. Zelinka, Comparison of various distances between isomorphism classes of graphs. Casopis Pést. Mat. 110 (1985) 289-293.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. THE GRADUATE COLLEGE WESTERN MICHIGAN UNIVERSITY KALAMAZOO, MICHIGAN

n . t . May 23. 1991

WE HEREBY APPROVE THE DISSERTATION SUBMITTED BY

ELZBIETA B. JARRETT

ENTITLED TRANSFORMATIONS OF GRAPHS AND DIGRAPHS

AS PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF. Doctor of Philosophy

Mathematics and Statistics (Department) Dissertation Review Committee Chair

0^.

Dissertation Review Committee Member

DissertatioiyReview Committee Member

Dissertation Review Committee Member

-Ç . APPROVED

Date_ /9 f/ Dean of The Graduate College

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.