Minimal Cycle Bases of Outerplanar Graphs
Josef Leydold Peter F. Stadler
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SANTA FE INSTITUTE
Minimal Cycle Bases of Outerplanar Graphs
a b c
Josef Leydold and Peter F Stadler
a
Dept for Applied Statistics and Data Pro cessing
University of Economics and Business Adminstration
Augasse A Wien Austria
Phone Fax E Mail Josef Leydold wu wien ac at
URL http statistik wu wien ac at staff leydold
b
Institut fur Theoretische Chemie Universit at Wien
W ahringerstra e A Wien Austria
Phone Fax E Mail studla tbi univie ac at
Address for corresp ondence
c
The Santa Fe Institute
Hyde Park Road Santa Fe NM USA
Phone Fax E Mail stadler santafe edu
URL http www tbi univie ac at studla
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 Classi cation C D
J Leydold P F Stadler Minimal Cycle Bases of Outerplanar Graphs
Intro duction
The description of cyclic structures is an imp ortant problem in graph theory
see e g Cycle bases of graphs have a variety of applications in science
and engineering 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 oly
mers 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 well de ned three dimen
sional structures These are of utmost imp ortance for their biological function
The most salien t 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 simpli cation 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 byavariety
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 sub cubic and outer planar
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 be computed as the sum of energy contributions of
the lo ops These lo ops form the unique minimal cycle basis of the contact graph
Exp 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 G V E with vertex set
V and edge set E i e there are no lo ops or multiple edges G V E is connected
if the deletion of a single vertex do es not disconnect the graph
Let G V E andG V E betwo sub graphs of a graph G V E We shall
write G n G for the subgraph of G induced bythe edge set E n E
The set E of all subsets of E forms an m dimensional vector space over GF
ultiplication X X with vector addition X Y X Y n X Y and scalar m
X for all X Y E A cycle is a subgraph such that anyvertex 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 G V E The set C of all cycles forms a subspace of
C
E which is called the cycle space of G A basis B of the cycle space C is called
a cycle basis of G V E The dimension of the cycle space is the cyclomatic
number or rst Betti number G jE j jV j
It is obvious that the cycle space of graph is the direct sum of the cycle spaces
of its connected comp onents It will be su cient therefore to consider only
connected graphs in this contribution
A connected or elementary cycle is a cycle C for which V C is a connected
C
minimal subgraph suchthatevery vertex in V has degree Wesay that a cycle C
J Leydold P F Stadler Minimal Cycle Bases of Outerplanar Graphs
basis is connected if all cycles are connected A cycle C is a chord less cycle if
V C is an induced subgraph of G V E i e if there is no edge in E n C that is
C
incidenttotwovertices of V We shall say that a cycle basis is chordless if all its
C
cycles are chordless
The length jC j of a cycle C is the numberofitsedges The length B of a cycle
P
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 c B b e the length of the longest cycle in
the cycle base B Chickering showed that B is minimal if and only if c B is
minimal i e a cycle basis is minimal if and only if has a shortest longest cycle
A cycle C is relevant ifitiscontained in a minimal cycle basis Vismara
proved the following
Prop ostion A cycle C is relevant if and only if it cannot be represented as a
sum of shorter cycles
An immediate consequence is
Corollary Arelevant cycle is chordless Hence a minimal cycle basis is chord
less and of course connected
Fundamental Cycle Bases
In what follows let G V E b e a connected graph
Supp ose T is a spanning tree of G Then for eachedge T there is unique
cycle in T f g 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 w r t T For details see A collection of G cycles in
G is called fundamental if there exists an ordering of these cycles suchthat
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 beordered 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 wehave G i i e 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
i
Otherwise there exists a j such connected for all i i e C G
i i
that C G for all C BnB for all orderings satisfying But then
j j
C C has emptyintersection with G C C a contradiction
j j j
G
since G C C is connected G is connected since by assumption all
i
G
C are connected
j
An immediate consequence is that C n G must b e either a path as claimed
j j
or an elementary cycle with has one vertex in common with G Otherwise we
j
G j If C n G is a cycle this one vertex must b e a cut vertex would have
j j j
of G Then there must be a cycle C BnB which has at least one edge in
j k j
common with G and with P Otherwise G cannot be connected Then we
j j
can reorder the basis by exchanging C and C
j k
Aweaker result holds for non fundamental cycle bases
J Leydold P F Stadler Minimal Cycle Bases of Outerplanar Graphs
Lemma Any connected non fundamental cycle basis can be ordered such that
C C is connected for al l i
i
Proof Analogously to the pro of of lemma second part we can show that all
G C C are connected
i i
If B is a non fundamental cycle basis of G then there is subgraph G with cycle
basis B B such that eachedgeofG is contained in at least two cycles of B
prop Furthermore the examples of non fundamental 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
veri ed 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 checkthat the following cycles are indep endent and thus are a basis of the
cycle space since K
Here denotes the cycle f v v v v v v g This basis is minimal
since every cycle has length But it is non fundamental since every edge is cov ered
at least two times
Outerplanar Graphs
A graph G V E isouter planar if it can b e emb edded in the plane such that all
vertices lie on the b oundary of its exterior region Given suchanemb edding wewill
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 invariantis discussed in
Lemma An outerplanar graph G V 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 cut edge
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 cut vertex of G since
it partitions B into at least two edge disjoint 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 nfxg with a vertex
z V nfxg Thus x is a cut vertex contradicting connectedness
Lemma A connected outerplanar graph G V E contains a unique Hamilton
ian 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
i
V V fp q g Now consider twovertices x V nfp q g and y V nfp q g Since
J Leydold P F Stadler Minimal Cycle Bases of Outerplanar Graphs
G is outerplanar each path from x to y passes through p or q Each connected
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 be part of any connected cycle containing both x and y Thus G
contains no Hamiltonian cycle di erent 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
n and n Without lo osing generalitywemay 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 i e the
set of contacts that are not interior to any other contact Yan s bambooshoot
graphs are exactly those outer planar graphs for which K is an ordered set
Nucleic acids both RNA and DNA form a sp ecial typ e of contact structure
known as sec ondary structure A graph G V E with V f ng is a secondary
structure if it satis es
i The so called backb one T f i i j i ng is a subset of E
ii For each i V there is at most one contact E n T incidentwithi
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 Agraph G V E isasecondary structuregraph is connected outerpla
nar 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 maydraw the chord E n T in the interior of this circle without intersection i e
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 be a connected outerplanar graph The set T H nf n g is
a spanning tree of G V E The fundamental basis F of the cycle space w r t T
therefore consists of the uniquely determined cycles F in T f g K and the
Hamiltonian cycle H T f n g We de ne
M M
H C F F and C F
Y Y
Furthermore we set M fC j K g fC g
J Leydold P F Stadler Minimal Cycle Bases of Outerplanar Graphs
Theorem Let G V E be a connected and outerplanar graph Then M is the
unique minimal cycle basis of G
Proof Consider an edge K suchthat Y that is a minimal element of the
p oset K We observe that F C in this case
Let G be the graph obtained from G by deleting the edges F nf g 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 nf g The fundamental basis F of G w r t
T H nf n g consists of H and the cycles F K which are obtained
by the rule F F C if and F F if Furthermore wehave
Y Y n and F C if and only if Y
Consider an arbitrary cycle basis B of G We can construct a cycle base 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 de ne Z Z if Z C or if Z C f g otherwise we
set Z Z C Z n C f g Wehave 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 i e B B
iii C B Then there is at least one cycle Z Z and all Z are non empty
Since C is indep endentofallZ there must b e at least on dep endent cycle
in the set fZ jZ Bg whichmust 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
That is B is strictly shorter than B in this case as well
Thus if B is a minimal cycle basis of G then cases ii and iii cannot o ccur i e
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 eachcycleC 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 G V E b e a planar graph and let fQ g b e the collection of faces in a given
j
emb edding in the plane EachfaceQ uniquely de nes the cycle Q which forms
j j
its b oundary The collection of cycles fQ g j G is a cycle basis of G
j
Any cycle basis obtained in this wayiscalleda planar cycle basis
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
Corollary M is planar cycle basis with length M jE j jV j
Proof The cycle basis M is the planar basis obtained byemb edding G in sucha
way in the plane that the Hamiltonian cycle H b ecomes the outer b oundary By
Using jH j jV j and jK j jE j jV j construction wehave M jH j jK j leads to the desired result
J Leydold P F Stadler Minimal Cycle Bases of Outerplanar Graphs
Figure Hamiltonian planar graph with a non planar minimal
cycle basis It is easy to verify that this graph has no planar em
b edding with the face Q A minimal cycle basis con
tains Q and two of the cycles
and Hence M while the planar bases have
M
In the following we state an algorithm for nding this unique minimal cycle
our investigation is motivated by RNA sec basis for a outerplanar graph Since
ondary structures we assume that the backb one of the outerplanar graph i e the
Hamiltonian cycle is already given Algorithm uses corollary to compute the
minimal cycle base As can easily b e seen this algorithm is of order O jV j To
show that it really delivers the planar and unique minimal cycle basis one can
draw the Hamiltonian path f ng as straight line add all contacts and edge
n above this line in suchaway that no twoofthemhave common p oints not
on the line
algorithm nd minimal cycle basis of outerplanar graphs
Input adjacency matrix Hamiltonian cycle f ng
for all vertices i i n do
k i
for all edges i k i k k do
j
pop edge i k fromstack This do es not work if G is not outerplanar
j
if k i then
add cycle f i i i i k k k i g to cycle basis
j j j
else
add cycle f i k k k k k k i g to cycle basis
j j j
k k
j
for all edges i k n k k i do
j
push edge i k onstack
j
It is interesting to note that the fundamental basis F can b e easily expressed in
terms of the minimal cycle basis M
ii h i h h
L L L
F F C F C C
Y Y Y
h h h iii
L L L
C F C C
Y Y Y
J Leydold P F Stadler Minimal Cycle Bases of Outerplanar Graphs
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 wehave
M
F C
W f g
Analogously one nds
M M
F C C C 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 G V 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
First we need the following simple observations
Prop ostion Let G E V be a connectedgraph 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 Euler s formula for
p olyhedra The upp er b ound on outerplanar graphs follo ws from a theorem by
G A Dirac stating that for any graph not containing K as a minor we have
jE j jV j
Abamb o o sho ot graph consisting of n triangles has n n vertices and
n n edges Consider the graph G recursively obtained byaddinga
n
vertex n which is connected to the three vertices lab eled n and of G
n
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
n n n
and 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 number of vertices
Theorem Let G E V be a connected planar graph with a minimal cycle
base M Then
M jV j if G is planar
M jV j if G is outerplanar
Proof Analogously to the pro of of lemma we nd for the planar case M
V j jV j by as claimed Similarly for the outer jE j j
planar case M jE j jV j jV j jV j jV j whichis
J Leydold P F Stadler Minimal Cycle Bases of Outerplanar Graphs
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
m
the example in gure Similar examples for jV j can b e constructed by
the following recip e
m
Makea gon
m
Insert center as additional vertex c and add edges c i for i
Insert edges
Insert edges and so on
Lemma Let G V 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 t wice 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 wehave
Corollary Every minimal cycle basis of a planar graph is fundamental
Proof Supp ose B is not fundamental By prop we can assume that every
edge is covered by at least two edges Thus B jE j But since every planar
cycle basis has length B jE j by lemma the prop osition follows
Lower Bounds on B
Theorem Let G V E bea connectedgraph and B a cycle basis of G Then
B jE j jV j
Equality holds for a minimal cycle basis B if and only if for every vertex v V the
number of cycles c B through v is d where d denotes the degreeof v
v v
Proof Let S denote the graph induced by all edges incident to v S is a star
v v
of diameter with d edges The edge set E of S with the addition forms
v v v
a d dimensional vector space Let S denote its sub space where each element
v v
consists of an even numb er of edges As can easily b e veri ed dim S d
v
Let C denote the vector space fC S C Eg It is obvious that C S
v v v v
Moreover for all C C that consists of exactly edges there exists a C E with
v
C C S and thus C S Otherwise every cycle C in G has vertex v as double
v v v
point and hence v was a cut vertex of G a contradiction to G b eing connected
J Leydold P F Stadler Minimal Cycle Bases of Outerplanar Graphs
Therefore C S is spanned by B fC S C B C S g Conse
v v v v v
quently jB j dim S d and B d since every C B contains at
v v v v
P
least two edges Notice that jE j d every edge is incidentto vertices
v
v V
P
B every edge is contained in the stars S and B centered at two
v v
i
v V
vertices v and v Hence we nd
X X
B d jE j jV j B
v v
v V v V
i e inequality Equality holds if and only if B is a basis of S Thus the
v v
statement follows
Theorem Let G V E be a connectedgraph with a minimum 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 i e consists of exactly one edge
i i
Proof By corollary B is elementary By lemmata and we can order the cycle
basis suchthat C C is connected Thus C C C consists of
i i i
at least one edge
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
i i i i i i i i
numb er of connected comp onents i e paths in C n E Notice that the number
i i
of comp onents in C E if and otherwise since C is elementary
i i i i i
Thus jE C j and
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 Wehave B jC j jE j jV j G sinceC
i i
is an elementary cycle By induction wethen nd B G jE j jV j
i i i
B B jC j G jC j G
i i i i i i
Equality holds if and only if all E C If B is fundamental then
i i i i
for all i by lemma and E C is a single edge for all i as claimed If B
i i
is not fundamental than we always have an j such that C E and thus
j j
jE C j jC j Thus B G
j j j j j j
In the following we derivesomeweaker conditions for which is sharp
Lemma Let G V E be a connectedgraph If B jE j jV j for a cycle
base B then G is planar
Proof If equality holds in equ then B is elementary and fundamental bytheo
rem Then 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 by in
i i
duction we can add the ear P from C into the planar drawing of C C
i i i
for all i
J Leydold P F Stadler Minimal Cycle Bases of Outerplanar Graphs
Lemma Let G V 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
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 be ordered such that every cycle C has exactly
i
one edge in common with C C Obviously H C is Hamiltonian cycle in
i
C Then by induction H H C is a Hamiltonian cycle in C C
i i i i
Furthermore we can draw the ears P of the cycles C lemma into the
i i
outside of C C Thus H is the b oundary to the exterior region of G and the
i i i
prop osition follows by induction
Figure Non Hamiltonian planar graph for which equality holds
in lemma
The condition Hamiltonian in lemma cannot b e relaxed An example of a
planar non Hamiltonian graph with B jE j jV j is shown in gure
References
G Chartrand and F Harary Planar permutation graphs Ann Inst Henri Poincar e B
W K Chen On vector spaces asso ciated with a graph SIAM J Appl Math
D M Chickering D Geiger and D Heckerman On nding a cycle basis of with a shortest
maximal cycle Inform Processing Let
Y Colin de Verdi ere Sur un novel invariant des graphes et un crit ere de planarit e J Comb
Theory B
N Deo G M Prabhu and M S Krishnamo orty Algorithms for generating fundamental
cycles in a graph ACM Trans Math Software
G A Dirac In abstrakten Graphen vorhandene vollst andige Graphen und ihre Unterteilun
gen Math Nachr
G M Downs V J Gillet J D Holliday and M F Lynch Review of ring p erception
algorithms for chemical graphs J Chem Inf Comput Sci
S M Freier R Kierzek J A Jaeger N Sugimoto M H Caruthers T Neilson and D H
Turner Improved free energy parameters for predictions of RNA duplex stability Proc Natl
Acad Sci USA
D Hartvigsen and E Zemel Is every cycle basis fundamental J Graph Theory
J D Horton A p olynomial time algorithm to nd the shortest cycle basis of a graph SIAM
J Comput
J Leydold P F Stadler Minimal Cycle Bases of Outerplanar Graphs
A Kaveh Structural Mechanics Graph and Matrix Methods Research Studies Press Exeter
UK
M Plotkin Mathematical basis of ring nding algorithms in CIDS J Chem Doc
K Thulasiraman and M N S Swamy Graphs Theory and Algorithms J Wiley Sons
New York
P Vismara Union of all the minimum cycle bases of a graph Electronic J Comb R
pages
H J Voss Cycles and Bridges in Graphs Kluwer Dordrecht
M S Waterman Secondary structure of single stranded nucleic acids Adv Math Suppl
Studies
H Whitney On abstract prop erties of linear dep endence Am J Math
L Yan A family of sp ecial outerplanar graphs with only one triangle satisfying the cycle
basis interp olation prop erty Discr Math