Minimal Cycle Bases of Outerplanar Graphs Josef Leydold Peter F. Stadler SFI WORKING PAPER: 1998-01-011 SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views of the Santa Fe Institute. We accept papers intended for publication in peer-reviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our external faculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, or funded by an SFI grant. ©NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensure timely distribution of the scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the author(s). It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may be reposted only with the explicit permission of the copyright holder. www.santafe.edu SANTA FE INSTITUTE Minimal Cycle Bases of Outerplanar Graphs a bc 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 EMail JosefLeydoldwuwienacat URL httpstatistikwuwienacatstaffleydold b Institut fur Theoretische Chemie Universitat Wien Wahringerstrae A Wien Austria Phone Fax EMail studlatbiunivieacat Address for corresp ondence c The Santa Fe Institute Hyde Park Road Santa Fe NM USA Phone Fax EMail stadlersantafeedu URL httpwwwtbiunivieacatstudla 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 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 eg 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 welldened 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 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 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 outerplanar 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 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 andG V E betwo subgraphs of a graph GV 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 mdimensional 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 GV 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 GV E The dimension of the cycle space is the cyclomatic number or rst Betti number GjE jjV 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 sucient 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 GV E ie 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 cB b e the length of the longest cycle in the cycle base B Chickering showed that B is minimal if and only if cB is minimal ie 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 GV 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 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 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 iieB 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 ie 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 nonfundamental cycle bases J Leydold P F Stadler Minimal Cycle Bases of Outerplanar Graphs Lemma Any connected nonfundamental 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 nonfundamental 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 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 checkthat 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 cov ered at least two times Outerplanar Graphs A graph GV Eisouterplanar 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