THE COMPLEXITY OF GREEDOID TUTTE POLYNOMIALS A thesis submitted for the degree of Doctor of Philosophy by Christopher N. Knapp Department of Mathematics, Brunel University London ABSTRACT We consider the computational complexity of evaluating the Tutte polynomial of three particular classes of greedoid, namely rooted graphs, rooted digraphs and binary greedoids. Furthermore we construct polynomial-time algorithms to evaluate the Tutte polynomial of these classes of greedoid when they’re of bounded tree-width. We also construct a M¨obius function formulation for the char- acteristic polynomial of a rooted graph and determine the computational complexity of computing the coefficients of the Tutte polynomial of a rooted graph. 1 ACKNOWLEDGEMENTS I would like to express my most sincere gratitude to my supervisor, Steve Noble. This thesis is a reflection of his expertise, innovation and progressive ideas. Not only this, but it also reflects the many hours he spent tirelessly guiding me with the upmost patience. His charisma and sense of humour both contributed to making our meetings so engaging, and I will be forever indebted to him for having confidence in me even when I didn’t have it in myself. I would also like to thank my second supervisor, Rhiannon Hall, for her much-needed support and constructive criticism, and the administrative staff in the department, in particular Frances Foster and Linda Campbell, for going above and beyond to answer all of my daily queries. I dedicate this thesis to my two biggest supporters - my mum and my nana, both of whom I sadly lost in 2016. 2 Contents Abstract 1 Acknowledgements 2 1 Introduction 5 1.1 Introduction...................................... ... 5 1.2 GraphsandMatroids................................. ... 6 1.3 RootedGraphs,RootedDigraphsandGreedoids . ........ 10 1.4 Complexity ......................................... 27 1.5 TheTuttePolynomialofaGraph. .... 30 1.6 The Complexity of Computing the Tutte Polynomial of a Graph . ....... 32 1.7 The Tutte Polynomial of an Arbitrary Greedoid, a Rooted Graph and a Rooted Digraph 33 2 The Computational Complexity of Evaluating the Tutte Polynomial at a Fixed Point 39 2.1 Introduction...................................... ... 39 2.2 GreedoidConstructions .............................. .... 40 2.3 RootedGraphs ..................................... .. 47 2.3.1 Introduction .................................... 47 2.3.2 ProofofMainTheorem .............................. 48 2.4 RootedDigraphs .................................... .. 61 2.5 BinaryGreedoids.................................... .. 73 3 Polynomial-Time Algorithms for Evaluating the Tutte Polynomial 77 3.1 Introduction...................................... ... 77 3 3.2 RootedGraphs ..................................... .. 83 3.2.1 Complexity of the Algorithm . 97 3.3 RootedDigraphs .................................... 100 3.3.1 Complexity of the Algorithm . 109 3.4 BinaryGreedoids.................................... 110 4 The Characteristic Polynomial and the Computational Complexity of the Coef- ficientsoftheTuttePolynomialofaRootedGraph 117 4.1 The Characteristic Polynomial . .... 117 4.2 TheCoefficients..................................... 128 4.2.1 Coefficients of T (G; x, 0).............................. 135 4.3 A Convolution Formula for the Tutte Polynomial of an Interval Greedoid . 139 4 Chapter 1 Introduction 1.1 Introduction This thesis comprises four chapters. Chapter 1 contains the foundations for the thesis. Two combina- torial structures, namely matroids and their generalization greedoids, are discussed and relationships between particular classes of them are explored. We discuss the fundamental ideas of computational complexity theory and delve into the wide range of results surrounding the renowned Tutte polyno- mial. Evaluating the Tutte polynomial of a graph at most fixed rational points is shown to be #P- hard in [31]. The k-thickening operation plays a significant role in the proof of this result. We begin Chapter 2 by generalizing the k-thickening operation to greedoids. We introduce two more greedoid constructions and give expressions for the Tutte polynomials resulting from these constructions. We then use these to find analogous results to those in [31] by completely determining the computational complexity of evaluating the Tutte polynomial of a rooted graph, a rooted digraph and of a binary greedoid at a fixed rational point. For the rooted graph case we also strengthen our results by restricting the problem to cubic, planar, bipartite rooted graphs. Many computational problems that are intractable for arbitrary graphs become easy for the class of graphs of bounded tree-width. In [42] Noble illustrates this result by constructing a polynomial- time algorithm to evaluate the Tutte polynomial of a graph of bounded tree-width. In Chapter 3 we construct polynomial-time algorithms to evaluate the Tutte polynomials of the classes considered in the previous chapter when each is of bounded tree-width. 5 The characteristic polynomial of a matroid is a specialization of the Tutte polynomial. In [64] Zaslavsky gives an expression for the characteristic polynomial of a matroid in terms of the M¨obius function. Gordon and McMahon generalize the characteristic polynomial to greedoids in [22] and show that several of the matroidal results have direct greedoid analogues. In Chapter 4 we begin by constructing a M¨obius function formulation for the characteristic polynomial of a rooted graph. In [2] Annan determines the computational complexity of computing the coefficients of the Tutte polynomial of a graph. He shows that for fixed non-negative integers i and j, computing the coefficient of xiyj in the Tutte polynomial of a graph is #P-hard. We continue Chapter 4 by finding analogous results for the coefficients of the Tutte polynomial of a rooted graph. We then study the smallest integer i such that the coefficient of xi in the Tutte polynomial expansion of a rooted graph is non-zero. In [34] Kook et al present a convolution formula for the Tutte polynomial of a matroid. A natural question would be to ask if we could extend this convolution formula to greedoids. We conclude Chapter 4 by conjecturing that if the conditions hold in the proof of this convolution formula for an interval greedoid then that interval greedoid is a matroid. 1.2 Graphs and Matroids We assume familiarity with the basic ideas of graph theory. For a more in-depth introduction to graphs, the reader is referred to [14] or [24], for example, from where the following definitions and basic results are taken. A graph G is a pair of sets (V (G), E(G)) where the elements of V (G) and E(G) are vertices and edges of G respectively. We can omit the argument when there is no fear of ambiguity. For the following graph definitions assume G = (V, E) and let A E and S V . Two vertices of a graph ⊆ ⊆ are adjacent if there exists an edge between them. Similarly two edges are adjacent if they share a common endpoint. An edge is incident to a vertex if the vertex is an endpoint of the edge. A set of edges are in parallel if they all share the same endpoints. The parallel class of an edge e E is the ∈ maximal set of edges that are in parallel with e in G, including e itself. The multiplicity of e is the cardinality of its parallel class, denoted by m(e). A set of pairwise non-adjacent vertices is said to be independent. A subgraph of G is a graph G′ = (V ′, E′) with V ′ V and E′ E. A subgraph G′ ⊆ ⊆ of G is spanning if V (G′)= V (G). The graph obtained by restricting the edges to those in A is given by G A = (V, A). The graphs | 6 obtained by deleting the sets A and S are given by G A = (V, E A) and G S = (V S, E E′) \ − \ − − respectively such that E′ is the set of edges incident to any vertex in S. If e is an edge of a graph G, then the graph G/e, the contraction of e from G, is obtained by removing e and identifying its endpoints into a single new vertex. If e and f are edges of G, then G/e/f = G/f/e, so the definition of contraction may be generalized to sets of edges without any ambiguity. Moreover, G/e f = G f/e, so we define a minor of G to be any graph obtained from G by deleting and \ \ contracting edges. The order of the deletions and contractions does not matter. We say G is connected if there is a path between any two vertices in G. A component of a graph is a maximal connected subgraph. We denote the number of components of G A by κ (A) and let | G κ(G)= κ (E(G)). The rank of A, denoted by r (A), is given by r (A)= V (G) κ (A). We let G G G | |− G r(G)= rG(E(G)). We can omit the subscripts in κG(A) and rG(A) when the context is clear. A graph is planar if it can be embedded in the plane without any of its edges crossing, and it is a plane graph if it is embedded in the plane in such a way. Every plane connected graph G has a dual graph G∗ formed by assigning a vertex of G∗ to each face of G, and j edges between vertices in G∗ whenever the corresponding faces of G share j edges in their boundaries. For a plane non- connected graph G we define the dual G∗ to be the union of the duals of each connected component. Clearly duality is an involution, that is (G∗)∗ = G for all G. If G is plane and A E(G) we have ⊆ r ∗ (A)= A r(G)+ r (E A). A more generalized variation of this result is proven in [62]. G | |− G − A loop of a graph is an edge whose endpoints are the same vertex, and a coloop, otherwise known as a bridge or an isthmus, is an edge whose deletion would increase the number of components. Thus a coloop is an edge that is not contained in any cycle.