Maximum Weight Independent Sets in Odd-Hole-Free Graphs Without Dart or Without Bull

The Maximum Weight Independent Set (MWIS) Problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. Being one of the most investigated and most important problems on graphs, it is well known to be NP-complete and hard to approximate. The complexity of MWIS is open for hole-free graphs (i.e., graphs without induced subgraphs isomorphic to a chordless cycle of length at least five). By applying clique separator decomposition as well as modular decomposition, we obtain polynomial time solutions of MWIS for odd-hole- and dart-free graphs as well as for odd-hole- and bull-free graphs (dart and bull have five vertices, say $a,b,c,d,e$, and dart has edges $ab,ac,ad,bd,cd,de$, while bull has edges $ab,bc,cd,be,ce$). If the graphs are hole-free instead of odd-hole-free then stronger structural results and better time bounds are obtained

Homogeneous sets, clique-separators, critical graphs, and optimal χ\chi-binding functions

Given a set $\mathcal{H}$ of graphs, let $f_\mathcal{H}^\star\colon \mathbb{N}_{>0}\to \mathbb{N}_{>0}$ be the optimal $\chi$-binding function of the class of $\mathcal{H}$-free graphs, that is, $$f_\mathcal{H}^\star(\omega)=\max\{\chi(G): G\text{ is } \mathcal{H}\text{-free, } \omega(G)=\omega\}.$$ In this paper, we combine the two decomposition methods by homogeneous sets and clique-separators in order to determine optimal $\chi$-binding functions for subclasses of $P_5$-free graphs and of $(C_5,C_7,\ldots)$-free graphs. In particular, we prove the following for each $\omega\geq 1$: (i) $\ f_{\{P_5,banner\}}^\star(\omega)=f_{3K_1}^\star(\omega)\in \Theta(\omega^2/\log(\omega)),$ (ii) $\ f_{\{P_5,co-banner\}}^\star(\omega)=f^\star_{\{2K_2\}}(\omega)\in\mathcal{O}(\omega^2),$(iii)$\ f_{\{C_5,C_7,\ldots,banner\}}^\star(\omega)=f^\star_{\{C_5,3K_1\}}(\omega)\notin \mathcal{O}(\omega),$and (iv)$\ f_{\{P_5,C_4\}}^\star(\omega)=\lceil(5\omega-1)/4\rceil.$We also characterise, for each of our considered graph classes, all graphs$G$with$\chi(G)>\chi(G-u)$for each$u\in V(G)$. From these structural results, we can prove Reed's conjecture -- relating chromatic number, clique number, and maximum degree of a graph -- for$(P_5,banner)$-free graphs

A polynomial Turing-kernel for weighted independent set in bull-free graphs

The maximum stable set problem is NP-hard, even when restricted to triangle-free graphs. In particular, one cannot expect a polynomial time algorithm deciding if a bull-free graph has a stable set of size k, when k is part of the instance. Our main result in this paper is to show the existence of an FPT algorithm when we parameterize the problem by the solution size k. A polynomial kernel is unlikely to exist for this problem. We show however that our problem has a polynomial size Turingkernel. More precisely, the hard cases are instances of size O(k5). As a byproduct, if we forbid odd holes in addition to the bull, we show the existence of a polynomial time algorithm for the stable set problem. We also prove that the chromatic number of a bull-free graph is bounded by a function of its clique number and the maximum chromatic number of its triangle-free induced subgraphs. All our results rely on a decomposition theorem for bull-free graphs due to Chudnovsky which is modified here, allowing us to provide extreme decompositions, adapted to our computational purpose

Colourings of P5P_5-free graphs

For a set of graphs H, we call a graph G H-free if G-S is non-isomorphic to H for each S⊆V(G) and each H∈H. Let f_H^* ∶N_(>0)↦N_(>0 )be the optimal χ-binding function of the class of H-free graphs, that is, f_H^* (ω)=max⁡{χ(G): ω(G)=ω,G is H-free} where χ(G),ω(G) denote the chromatic number and clique number of G, respectively. In this thesis, we mostly determine optimal χ-binding functions for subclasses of P_5-free graphs, where P_5 denotes the path on 5 vertices. For multiple subclasses we are able to determine them exactly and for others we prove the right order of magnitude. To achieve those results we prove structural results for the graph classes and determine colourings. We sometimes obtain those results by researching the prime graphs and combining the two decomposition methods by homogeneous sets and clique-separators. Additionally, we use the Strong Perfect Graph Theorem and analyse the neighbourhood of holes. For some of these subclasses we characterise all graphs G with χ(G)>χ(G-\{u\}), for each u∈V(G) and use those to determine the function

On Minimum Dominating Sets in cubic and (claw,H)-free graphs

Given a graph $G=(V,E)$, $S\subseteq V$ is a dominating set if every $v\in V\setminus S$ is adjacent to an element of $S$. The Minimum Dominating Set problem asks for a dominating set with minimum cardinality. It is well known that its decision version is $NP$-complete even when $G$ is a claw-free graph. We give a complexity dichotomy for the Minimum Dominating Set problem for the class of $(claw, H)$-free graphs when $H$ has at most six vertices. In an intermediate step we show that the Minimum Dominating Set problem is $NP$-complete for cubic graphs

Quantum Computation, Markov Chains and Combinatorial Optimisation

This thesis addresses two questions related to the title, Quantum Computation, Markov Chains and Combinatorial Optimisation. The first question involves an algorithmic primitive of quantum computation, quantum walks on graphs, and its relation to Markov Chains. Quantum walks have been shown in certain cases to mix faster than their classical counterparts. Lifted Markov chains, consisting of a Markov chain on an extended state space which is projected back down to the original state space, also show considerable speedups in mixing time. We design a lifted Markov chain that in some sense simulates any quantum walk. Concretely, we construct a lifted Markov chain on a connected graph G with n vertices that mixes exactly to the average mixing distribution of a quantum walk on G. Moreover, the mixing time of this chain is the diameter of G. We then consider practical consequences of this result. In the second part of this thesis we address a classic unsolved problem in combinatorial optimisation, graph isomorphism. A theorem of Kozen states that two graphs on n vertices are isomorphic if and only if there is a clique of size n in the weak modular product of the two graphs. Furthermore, a straightforward corollary of this theorem and Lovász’s sandwich theorem is if the weak modular product between two graphs is perfect, then checking if the graphs are isomorphic is polynomial in n. We enumerate the necessary and sufficient conditions for the weak modular product of two simple graphs to be perfect. Interesting cases include complete multipartite graphs and disjoint unions of cliques. We find that all perfect weak modular products have factors that fall into classes of graphs for which testing isomorphism is already known to be polynomial in the number of vertices. Open questions and further research directions are discussed