Subgraphs and Colourability of Locatable Graphs

We study a game of pursuit and evasion introduced by Seager in 2012, in which a cop searches the robber from outside the graph, using distance queries. A graph on which the cop wins is called locatable. In her original paper, Seager asked whether there exists a characterisation of the graph property of locatable graphs by either forbidden or forbidden induced subgraphs, both of which we answer in the negative. We then proceed to show that such a characterisation does exist for graphs of diameter at most 2, stating it explicitly, and note that this is not true for higher diameter. Exploring a different direction of topic, we also start research in the direction of colourability of locatable graphs, we also show that every locatable graph is 4-colourable, but not necessarily 3-colourable.Comment: 25 page

Vertex colouring and forbidden subgraphs - a survey

There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions

Cliques, colouring and satisfiability : from structure to algorithms

We examine the implications of various structural restrictions on the computational complexity of three central problems of theoretical computer science (colourability, independent set and satisfiability), and their relatives. All problems we study are generally NP-hard and they remain NP-hard under various restrictions. Finding the greatest possible restrictions under which a problem is computationally difficult is important for a number of reasons. Firstly, this can make it easier to establish the NP-hardness of new problems by allowing easier transformations. Secondly, this can help clarify the boundary between tractable and intractable instances of the problem. Typically an NP-hard graph problem admits an infinite sequence of narrowing families of graphs for which the problem remains NP-hard. We obtain a number of such results; each of these implies necessary conditions for polynomial-time solvability of the respective problem in restricted graph classes. We also identify a number of classes for which these conditions are sufficient and describe explicit algorithms that solve the problem in polynomial time in those classes. For the satisfiability problem we use the language of graph theory to discover the very first boundary property, i.e. a property that separates tractable and intractable instances of the problem. Whether this property is unique remains a big open problem

A refinement on the structure of vertex-critical (P5P_5, gem)-free graphs

We give a new, stronger proof that there are only finitely many $k$-vertex-critical ($P_5$,~gem)-free graphs for all $k$. Our proof further refines the structure of these graphs and allows for the implementation of a simple exhaustive computer search to completely list all $6$- and $7$-vertex-critical $(P_5$, gem)-free graphs. Our results imply the existence of polynomial-time certifying algorithms to decide the $k$-colourability of $(P_5$, gem)-free graphs for all $k$ where the certificate is either a $k$-colouring or a $(k+1)$-vertex-critical induced subgraph. Our complete lists for $k\le 7$ allow for the implementation of these algorithms for all $k\le 6$

Structural solutions to maximum independent set and related problems

In this thesis, we study some fundamental problems in algorithmic graph theory. Most natural problems in this area are hard from a computational point of view. However, many applications demand that we do solve such problems, even if they are intractable. There are a number of methods in which we can try to do this: 1) We may use an approximation algorithm if we do not necessarily require the best possible solution to a problem. 2) Heuristics can be applied and work well enough to be useful for many applications. 3) We can construct randomised algorithms for which the probability of failure is very small. 4) We may parameterize the problem in some way which limits its complexity. In other cases, we may also have some information about the structure of the instances of the problem we are trying to solve. If we are lucky, we may and that we can exploit this extra structure to find efficient ways to solve our problem. The question which arises is - How far must we restrict the structure of our graph to be able to solve our problem efficiently? In this thesis we study a number of problems, such as Maximum Indepen- dent Set, Maximum Induced Matching, Stable-II, Efficient Edge Domina- tion, Vertex Colouring and Dynamic Edge-Choosability. We try to solve problems on various hereditary classes of graphs and analyse the complexity of the resulting problem, both from a classical and parameterized point of view

Colouring on hereditary graph classes

The graph colouring problems ask if one can assign a colour from a palette of colour to every vertex of a graph so that any two adjacent vertices receive different colours. We call the resulting problem k-Colourability if the palette is of fixed size k, and Chromatic Number if the goal is to minimize the size of the palette. One of the earliest NP-completeness results states that 3-Colourability is NP-complete. Thereafter, numerous studies have been devoted to the graph colouring problems on special graph classes. For a fixed set of graphs H we denote F orb(H) by the set of graphs that exclude any graph H ∈ H as an induced subgraph. In this thesis, we explore the computational complexity of graph colouring problems on F orb(H) for different sets of H.In the first part of this thesis, we study k-Colourability on classes F orb(H) when H contains at most two graphs. We show that 4-Colourability and 5-Colourability are NPcomplete on F orb({P7}) and F orb({P6}), respectively, where Pt denotes a path of order t. These results leave open, for k ≥ 4, only the complexity of k-Colourability on F orb({Pt}) for k = 4 and t = 6. Secondly, we refine our NP-completeness results on k-Colourability to classes F orb({Cs, Pt}), where Cs denotes a cycle of length s. We prove new NP-completeness results for different combinations of values of k, s and t. Furthermore, we consider two common variants of the k-colouring problem, namely the list k-colouring problem and the pre-colouring extension of k-colouring problem. We show that in most cases these problems are also NP-complete on the class F orb({Cs, Pt}). Thirdly, we prove that the set of forbidden induced subgraph that characterizes the class of k-colourable (C4, P6)-free graphs is of finite size. For k ∈ {3, 4}, we obtain an explicit list of forbidden induced subgraphs and the first polynomial certifying algorithms for k-Colourability on F orb({C4, P6}).We also discuss one particular class F orb(H) when the size of H is infinite. We consider the intersection class of F orb({C4, C6, . . .}) and F orb(caps), where a cap is a graph obtained from an induced cycle by adding an additional vertex and making it adjacent to two adjacent vertices on the cycle. Our main result is a polynomial time 3/2-approximation algorithm for Chromatic Number on this class