Minimal Cycle Bases of Outerplanar Graphs

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Josef Leydold and Peter F. Stadler Minimal Cycle Bases of Outerplanar Graphs

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Leydold, Josef and Stadler, Peter F. (1998) Minimal Cycle Bases of Outerplanar Graphs. Preprint Series / Department of Applied Statistics and Data Processing, 23. Department of Statistics and Mathematics, Abt. f. Angewandte Statistik u. Datenverarbeitung, WU Vienna University of Economics and Business, Vienna. This version is available at: https://epub.wu.ac.at/528/ Available in ePubWU: July 2006 ePubWU, the institutional repository of the WU Vienna University of Economics and Business, is provided by the University Library and the IT-Services. The aim is to enable open access to the scholarly output of the WU.

http://epub.wu.ac.at/ Minimal Cycle Bases of Outerplanar Graphs

Josef Leydold, Peter F. Stadler

Department of Applied Statistics and Data Processing Wirtschaftsuniversitat¨ Wien

Preprint Series

Preprint 23 January 1998

http://statmath.wu-wien.ac.at/

Minimal Cycle Bases of Outerplanar Graphs

a bc

Josef Leydold and Peter F Stadler

Dept for Applied Statistics and Data Pro cessing

University of Economics and Business Administration

Augasse A Wien Austria

Phone Fax

EMail JosefLeydoldwuwienacat

URL httpstatistikwuwienacatstaffleydold

Institut fur Theoretische Chemie Universitat Wien

Wahringerstrae A Wien Austria

Phone Fax

EMail studlatbiunivieacat

Address for corresp ondence

The Santa Fe Institute

Hyde Park Road Santa Fe NM USA

Phone Fax

EMail stadlersantafeedu

URL httpwwwtbiunivieacatstudla

Submitted July Accepted February

Abstract

connected outerplanar graphs have a unique minimal cycle basis with length

jE j jV j They are the only Hamiltonian graphs with a cycle basis of this

length

Keywords Minimal Cycle Basis Outerplanar Graphs

AMS Sub ject Classication Primary C Secondary D 1

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Intro duction

The description of cyclic structures is an imp ortant problem in graph theory see

eg Cycle bases of graphs have a variety of applications in science and en

gineering among them in structural analysis and in chemical structure storage

and retrieval systems Naturally minimal cycles bases are of particular practical

interest

In this contribution we prove that outerplanar graphs have a unique minimal cycle

basis This result was motivated by the analysis of the structures of biop olymers In

addition we derive upp er and lower b ounds on the length of minimal cycle basis in

connected graphs

Biop olymers such as RNA DNA or proteins form welldened three dimensional

structures These are of utmost imp ortance for their biological function The most

salient features of these structures are captured by their contact graph representing the

set E of all pairs of monomers V that are spatially adjacent While this simplication

of the D shap e obviously neglects a wealth of structural details it encapsulates the

typ e of structural information that can b e obtained by a variety of exp erimental and

computational metho ds Nucleic acids b oth RNA and DNA form a sp ecial typ e of

contact structures known as secondary structures These graphs are outerplanar and

sub cubic ie the maximal vertex degree is

A particular typ e of cycles which is commonly termed lo ops in the RNA litera

ture plays an imp ortant role for RNA and DNA secondary structures the energy

of a secondary structure can b e computed as the sum of energy contributions of the

lo ops These lo ops form the unique minimal cycle basis of the contact graph Ex

p erimental energy parameters are available for the contribution of an individual lo op

as a function of its size of the typ e of b onds that are contained in it and on the

monomers nucleotides that it is comp osed of Based on this energy mo del it is

p ossible to compute the secondary structure with minimal energy given the sequence

of nucleotides using a dynamic programming technique

Preliminaries

In this contribution we consider only nite simple graphs GV E with vertex set

V and edge set E ie there are no lo ops or multiple edges GV E is connected

if the deletion of a single vertex do es not disconnect the graph

Let G V E and G V E b e two subgraphs of a graph GV E We shall

write G n G for the subgraph of G induced by the edge set E n E

The set E of all subsets of E forms an mdimensional vector space over GF with

vector addition X Y X Y n X Y and scalar multiplication X X

X for all X Y E A cycle is a subgraph such that any vertex degree is even

We represent a cycle by its edge set C Sometimes it will b e convenient to regard C

as a subgraph V C of GV E The set C of all cycles forms a subspace of E

which is called the cycle space of G A basis B of the cycle space C is called a cycle

basis of GV E The dimension of the cycle space is the cyclomatic number or

rst Betti number G jE j jV j

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It is obvious that the cycle space of graph is the direct sum of the cycle spaces of its

connected comp onents It will b e sucient therefore to consider only connected

graphs in this contribution

A elementary cycle is a cycle C for which V C is a connected minimal subgraph

such that every vertex in V has degree We say that a cycle basis is elementary

if all cycles are elementary A cycle C is a chord less cycle if V C is an induced

subgraph of GV E ie if there is no edge in E n C that is incident to two vertices

of V We shall say that a cycle basis is chordless if all its cycles are chordless

The length jC j of a cycle C is the numb er of its edges The length B of a cycle

basis B is sum of the lengths of its cycles B jC j A minimal cycle basis

C B

is a cycle basis with minimal length Let cB b e the length of the longest cycle in

the cycle basis B Chickering showed that if B is minimal then cB is minimal

ie a minimal cycle basis has a shortest longest cycle

A cycle C is relevant if it is contained in a minimal cycle basis Vismara

proved the following

Prop osition A cycle C is relevant if and only if it cannot be represented as a

sum of shorter cycles

An immediate consequence is

Corollary A relevant cycle is chord less Hence a minimal cycle basis is chord less

and of course elementary

Fundamental Cycle Bases

In what follows let GV E b e a connected graph

A collection of G cycles in G is called fundamental if there exists an ordering of

these cycles such that

C n C C C for j G

j j

Of course such a collection is a cycle basis Not all cycle bases are fundamental

Lemma An elementary fundamental cycle basis can be ordered such that

i C is an elementary cycle and

ii C n C C P is a nonempty path for j G

j j j

Proof Let G C C Then G G for i and consequently

i i i i

G G G G G G Therefore

G G

equality holds and we have G i ie B fC C g is a cycle basis for G

i i i i

Next notice that there exists an ordering for which holds such that G is con

nected for all i ie C G Otherwise there exists a j such that C G

i i j

for all C B n B for all orderings satisfying But then C C has empty

j j

intersection with G C C a contradiction since G C C

j j

is connected G is connected since by assumption all C are elementary

i j

An immediate consequence is that C n G must b e either a path as claimed or

j j

an elementary cycle which has one vertex in common with G Otherwise we would

have G j If C n G is a cycle this one vertex must b e a cut vertex of G In

j j j j

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this case there must b e a list of cycles C C C in B n B such that C has

k k k j k

1 2 1

edges in common with G C has edges in common with C for all q p

j k k

q +1

and C has edges in common with C Then we can reorder the basis by exchanging

k j

C and C

j k

A weaker result holds for nonfundamental cycle bases

Lemma Any elementary nonfundamental cycle basis can be ordered such that

C C is connected for al l i

Proof Analogously to the pro of of lemma there exists an ordering such that G

is connected for all i ie C G Otherwise there exists a j such that

i i

C G for all C B n B for all orderings But then C C has empty

j j j G

intersection with G C C a contradiction since G C C

j j G

is connected G is connected since by assumption all C are elementary

i j

Similarly there exists an ordering such that G is connected for all i Obvi

ously G C is connected since C is elementary Assume G is connected If

j j j

C G consists of at least two vertices G is connected If C and G have only

j j j j j

one vertex in common there must b e at least one such vertex then there must b e a

cycle C B n B which has edges in common with G and with P Otherwise G

k j j j

cannot b e connected Then we can reorder the basis by exchanging C and C

j k

If B is a nonfundamental cycle basis of G then there is subgraph G with cycle

basis B B such that each edge of G is contained in at least two cycles of B

prop Furthermore the examples of nonfundamental bases in are much

longer than the minimal cycles bases One might b e tempted therefore to conjecture

that every minimal cycle basis is fundamental Although this statement is easily

veried for planar graphs see corollary it is not true in general Consider the

complete graph K with vertices It is straightforward we used Mathematica to

check that the following cycles are indep endent and thus are a basis of the cycle

space since K

Here denotes the cycle fv v v v v v g This basis is minimal

since every cycle has length But it is nonfundamental since every edge is covered

at least two times

The concept of fundamental has originally b een intro duced by Kirchho in

in the following way

Supp ose T is a spanning tree of G Then for each edge T there is unique

cycle in T fg which is called a fundamental cycle The set of fundamental cycles

b elonging to a given spanning tree form a basis of the cycle subspace which is called

the fundamental basis wrt T For details see It is obvious that a fundamental

basis wrt a spanning tree is a sp ecial case of the fundamental collections dened at

the b eginning of this section see also

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Outerplanar Graphs

A graph GV E is outerplanar if it can b e emb edded in the plane such that all

vertices lie on the b oundary of its exterior region Given such an emb edding we will

refer to the set of edges on the b oundary to the exterior region as the boundary B of

G A graph is outerplanar if and only if it do es not contain a K or K minor

An algebraic characterization in terms of a sp ectral invariant is discussed in

Lemma An outerplanar graph GV E is Hamiltonian if and only if it is

connected

Proof A Hamiltonian graph is always connected Supp ose G is outerplanar

connected but not Hamiltonian connectedness implies that there is no cutvertex

Thus the b oundary B of G is a closed path containing an edge at most once A vertex

x that is incident with more than edges of B must b e a cutvertex of G since it

partitions B into at least two edgedisjoint closed paths B and B Let V and V

the vertices incident with the edges in B and B resp ectively Outerplanarity implies

that there are no edges connecting a vertex y V n fxg with a vertex z V n fxg

Thus x is a cut vertex contradicting connectedness

Lemma A connected outerplanar graph GV E contains a unique Hamiltonian

cycle H

Proof If G is a cycle graph there is nothing to show Otherwise denote by H the

Hamiltonian cycle forming the b oundary of G and consider an arbitrary edge H

By construction G is emb edded in the plane such that p q divides G into two

subgraphs G and G with vertex sets V and V satisfying jV j and V V fp q g

Now consider two vertices x V n fp q g and y V n fp q g Since G is outerplanar

each path from x to y passes through p or q Each elementary cycle containing b oth

x and y therefore consists of two disjoint paths one of which passes only through p

while the other one passes only through q Thus the edge p q cannot b e part of

any elementary cycle containing b oth x and y and hence G contains no Hamiltonian

cycle dierent from H

As a consequence there is a unique partition of the edge set E of an outerplanar

graph G into the Hamiltonian cycle H and the set of chords K E n H It will b e

convenient to lab el the vertices such that the edges in H are i i for i n

and n Without lo osing generality we may assume that n is a vertex of degree

It will b e useful to intro duce the following partial order on K

i j i j if and only if i i j j and

We say that is interior to If there is no K such that we say

that is immediately interior to For each alpha in K we set Y f

K j g Y denotes the set of maximal elements in K ie the set of contacts

that are not interior to any other contact Yans bamboo shoot graphs are exactly

those outerplanar graphs for which K is an ordered set

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Nucleic acids b oth RNA and DNA form a sp ecial typ e of contact structure known

as secondary structure A graph GV E with V f ng is a secondary struc

ture if it satises

i The socalled backb one T fi i j i ng is a subset of E

ii For each i V there is at most one contact E n T incident with i

iii If i j k l E n T and i k j then i l j

The contacts in nucleic acids are usually called base pairs Note that the backb one T

is a spanning tree of G

Lemma A secondary structure graph is connected outerplanar and subcubic

Proof By prop erties i and ii it is clear that a secondary structure graph is sub

cubic Prop erty iii implies that when the vertices are arranged along a circle then

one may draw the chord E n T in the interior of this circle without intersection ie

G is outerplanar This is a common representation for drawing RNA secondary

structures

The converse is not true since outerplanar sub cubic graphs do not necessarily have

unbranched spanning trees T

Minimal Cycle Bases of Outerplanar Graphs

Let G V b e a connected outerplanar graph The set T H n f ng is a

spanning tree of GV E The fundamental basis F of the cycle space wrt T in the

sense of Kirchho therefore consists of the uniquely determined cycles F in T fg

K and the Hamiltonian cycle H T f ng We dene

M M

H F F and C F C

Y Y

Furthermore we set M fC j K g fC g

Theorem Let GV E be a connected and outerplanar graph Then M is the

unique minimal cycle basis of G

Proof Consider an edge K such that Y that is a minimal element of the

p oset K We observe that F C in this case

Let G b e the graph obtained from G by deleting the edges F n fg and all

vertices that are isolated as a consequence It is clear that G is again a connected

outerplanar graph Its b oundary is the Hamiltonian cycle H H C The set of

chords of G is K K n fg The fundamental basis F of G wrt T H n f ng

consists of H and the cycles F K which are obtained by the rule F F C

if and F F if Furthermore we have Y Y n and F C if

and only if Y

Consider an arbitrary cycle basis B of G We can construct a cycle basis B of G

from B that consists of C and a cycle basis B of G by the following pro cedure For

each Z B we dene Z Z if Z C or if Z C fg In the remaining

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cases where Z C fg or b ecause Y we set Z Z C Z n C fg

We have to distinguish three cases

i C B and Z Z for all other cycles Then B B B fC g and

B B jC j

ii C B but there is at least one cycle Z B satisfying Z Z The length of

this cycle is jZ j jZ j jC j jZ j ie B B

iii C B Then there is at least one cycle Z Z and all Z are nonempty Since

C is indep endent of all Z there must b e at least on dep endent cycle in the set

fZ jZ B g which must b e removed in order to obtain the basis B The length

of this cycle is of course at least Thus

B B jC j jZ j jZ j B jC j jC j B

and B is strictly shorter than B in this case to o

Thus if B is a minimal cycle basis of G then cases ii and iii cannot o ccur ie a

minimal cycle basis of G consists of C and a minimal cycle basis B of G

Rep eating this argument jK j times shows that each cycle C K must b e

contained in any minimal cycle basis of G The remainder G of G after all cycles

C K are removed by the ab ove pro cedure is comp osed of Y and those edges of

H that are not contained in any of the cycles C The edge set of G is the chordless

cycle C Thus fC g fC j K g M is therefore the only minimal cycle basis of

Let GV E b e a planar graph and let f Q g b e the collection of faces in a given

emb edding in the plane Each face Q uniquely denes the cycle Q which forms its

j j

b oundary The collection of cycles fQ g j G is a cycle basis of G Any

cycle basis obtained in this way is called a planar cycle basis

Figure Hamiltonian planar graph with a nonplanar minimal cycle

basis It is easy to verify that this graph has no planar emb edding with

the face Q A minimal cycle basis contains Q and two

of the cycles and

Hence M while the planar bases have M

It is natural to ask whether every planar graph has a minimal cycle basis that is

also planar The answer to the question is negative in general as gure shows

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Corollary M is planar cycle basis with length M jE j jV j

Proof The cycle basis M is the planar basis obtained by emb edding G in such a

way in the plane that the Hamiltonian cycle H b ecomes the outer b oundary By

construction we have M jH j jK j Using jH j jV j and jK j jE j jV j leads

to the desired result

We now turn to an algorithm for nding the unique minimal cycle basis of an

outerplanar graph Since our investigation is motivated by RNA secondary structures

we assume that the backb one of the outerplanar graph ie the Hamiltoniancycle is

already given

The basic idea of algorithm is to nd the minimal cycles C describ ed in the pro of

of theorem When such a cycle is found it is added to the cycle basis and chopp ed

o the graph Step generates an ordered list of V along the Hamiltonian cycle

It is b est implemented as linked list of p ointers to the vertices Steps and push

every contact and n that have not already b een pro cessed on the stack Steps

and p op all contacts incident to the current vertex i from the stack By corollary

the ordering of the edges used in step ensures that the cycle generated in steps

and are chordless and all vertices except i and k in P have degree Hence they

are part of M In step step they are chops o taking advantage of the fact that

V is a linked list It is easy to see that this algorithm is of order O jV j We illustrate

the algorithm in Figure

It is interesting to note that the fundamental basis F can b e easily expressed in

terms of the minimal cycle basis M

ii i h h h

L L L

C C F C F F

Y Y Y

h iii h h

L L L

C C C F

Y Y Y

The expansion eventually stops if Y and hence F C Clearly the nested

sums contain each b ond in W f K j g the set of contacts interior to

and itself exactly once Therefore we have

F C

W fg

Analogously one nds

M M

C C C F H

K Y

Upp er Bounds on min `B

In theorem an upp er b ound for the length of a minimal cycle basis M of an

arbitrary graph GV E is given

M jV j jV j

While this b ound is sharp for complete graphs it can b e improved substantially

for planar graphs

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algorithm nd minimal cycle basis of outerplanar graphs

Input adjacency matrix Hamiltonian cycle f ng

V n

for all vertices i i n do

while there is an edge i k at the top of the stack do

p op edge i k from stack

P path from k to i in V

add cycle P fi k g to cycle basis

remove vertices in P n fi k g from V

for all edges i k n k k i do

push edge i k on stack

i step action b ottom stack top V

b eginn empty empty

make list empty

push

none

create

push

none

create

create empty

stop

Figure Example for algorithm

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First we need the following simple observations

Prop osition Let GE V be a connected graph Then

jE j jV j if G is planar

jE j jV j if G is outerplanar

These bounds are sharp for al l jV j

The result on planar graphs is an immediate corollary of Eulers formula for p oly

hedra The upp er b ound on outerplanar graphs follows from a theorem by GA Dirac

stating that for any graph not containing K as a minor we have jE j jV j

A bamb o osho ot graph consisting of n triangles has n n vertices and

n n edges Consider the graph G recursively obtained by adding a

vertex n which is connected to the three vertices lab eled n and of G We

set G K the cycle of length It is obvious that these graphs are all planar

and G has edges and vertex more than G Thus G has n vertices and

n n n

n n edges

We can translate the ab ove result into upp er b ounds for the lengths of a minimal

cycle bases that dep end only on the numb er of vertices

Theorem Let GE V be a connected planar graph with a minimal cycle basis

M Then

M jV j if G is planar

M jV j if G is outerplanar

Proof Analogously to the pro of of prop osition we nd for the planar case M

jE j jV j jV j by as claimed Similarly for the outer

planar case M jE j jV j jV j jV j jV j which is

Figure A Hamiltonian planar graph for which inequality is sharp

It is not p ossible to improve the b ound for planar Hamiltonian graphs see the

example in gure Similar examples for jV j can b e constructed by the

following recip e

Start with a gon

Insert the center as additional vertex c and add edges c i for i

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Insert edges

Insert edges and so on

Lemma Let GV E be a connected planar graph Then M jE j g G

where g G denotes the girth of G

Proof A planar cycle basis contains each interior edge twice while the edges of

the outer b oundary app ear only once The length of the outer b oundary is at least

g G

As an immediate consequence we have

Corollary Every minimal cycle basis of a planar graph is fundamental

Proof Supp ose B is not fundamental By prop we can assume that here is

a subset B B such that i B is a minimal cycle basis for G the subgraph of G

induced by B and ii B covers every edge of G two or more times We can assume

that G is connected otherwise consider G with minimal cycle basis B where G

is a connected subgraph of G with minimal cycle basis B B is a subset Note

that B necessarily covers every edge of G two or more times Thus B jE j

But since every planar cycle basis has length B jE j by lemma B cannot b e

a minimal cycle basis of G and the prop osition follows

Lower Bounds on `B

Theorem Let GV E be a connected graph and B a cycle basis of G Then

B jE j jV j

If equality holds then the cycle basis B is minimal Equality holds for a basis if and

only if for every vertex v V the number of cycles c B through v is d where

d denotes the degree of v

Proof Let S denote the graph induced by all edges incident to v S is a star of

v v

diameter with d edges The edge set E of S with the addition forms a d

v v v v

dimensional vector space Let T denote its sub space where each element consists of

an even numb er of edges As can easily b e veried dimT d

v v

Let C denote the vector space fC S C C g It is obvious that C T

v v v v

Moreover for all C C that consists of exactly edges there exists a C C with

C C S and thus C T Otherwise every cycle C in G has vertex v as double

v v v

p oint and hence v was a cut vertex of G a contradiction to G b eing connected

Therefore C T is spanned by B fC S C B C S g Consequently

v v v v v

jB j dimT d and B d since every C B contains at least

v v v v v

d every edge is incident to vertices and two edges Notice that jE j

v V

B centered at two vertices B every edge is contained in the stars S

v v

v V

v and v Hence we nd

X X

B d jE j jV j B

v v

v V v V

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ie inequality Equality holds if and only if B is a basis of T Thus the statement

v v

follows

Theorem Let GV E be a connected graph with a elementary cycle basis B

Then B jE j jV j if and only if B is a fundamental cycle basis such that

jC C C j for al l i G ie consists of exactly one edge In

i i

this case B is a minimal cycle basis

Proof By lemmata and we can order the cycle basis B such that C C is

connected Thus C C C consists of at least one edge

i i

Let G jE j jV j Let E denote the edge set of G and let V and V C

i i i j

denote the vertex sets of G and C resp ectively Then we nd

i j

G jE j jV j

i i i

jE j jC j jE C j jV j jV C j jV V C j

i i i i i i i i

jE j jV j jC j jV C j jE C j jV V C j

i i i i i i i i

G jC j jE C j

i i i i i

since jC j jV C j and jV V C j jE C j where denotes the numb er

i i i i i i i i

of connected comp onents ie paths in C n E The latter equality holds b ecause

i i

C is elementary Notice that the numb er of comp onents in C E if

i i i i i

and otherwise since C is elementary Thus jE C j and

i i i i

G G jC j

i i i

Equality holds if and only if jE C j

i i i

Let B fC C g We have B jC j jE j jV j G since C is

i i

an elementary cycle By induction we then nd B G jE j jV j

i i i i

B B jC j G jC j G

i i i i i i

Equality holds if and only if all jE C j If B is fundamental then for all

i i i i

i by lemma and E C is a single edge for all i as claimed If B is not fundamental

i i

than we always have an j such that C E and thus jE C j jC j

j j j j j j

Thus B G

j j

Moreover if B jE j jV j then by theorem B is a minimal cycle basis

In the following we derive some weaker conditions for which is sharp

Lemma Let GV E be a connected graph If B jE j jV j for a cycle

basis B then G is planar

Proof By theorems equality in equation implies that B is minimal elementary

and fundamental Thus there exists an ordering of B as describ ed in lemma By

theorem every cycle C has exactly one edge in common with C C Thus

i i

by induction we can add the ear P from C into the planar drawing of C C

i i i

for all i

Lemma Let GV E be a Hamiltonian graph Then there exists a cycle basis B

for which B jE j jV j holds if and only if G is outerplanar

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Proof By corollary a minimal cycle basis of an outerplanar graph has length jE j

jV j

If equality holds in equ then G is planar lemma and B is fundamental

theorem Moreover B can b e ordered such that every cycle C has exactly one

edge in common with C C Obviously H C is Hamiltonian cycle in C Then

by induction H H C is a Hamiltonian cycle in C C Furthermore we

i i i i

can draw the ears P of the cycles C lemma into the outside of C C

i i i

Thus H is the b oundary to the exterior region of G and the prop osition follows by

i i

induction

Figure NonHamiltonian planar graph for which equality holds in lemma

The condition Hamiltonian in lemma cannot b e relaxed An example of a

planar nonHamiltonian graph with B jE j jV j is shown in gure

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