28 research outputs found

    A Brightwell-Winkler type characterisation of NU graphs

    Full text link
    In 2000, Brightwell and Winkler characterised dismantlable graphs as the graphs HH for which the Hom-graph Hom(G,H){\rm Hom}(G,H), defined on the set of homomorphisms from GG to HH, is connected for all graphs GG. This shows that the reconfiguration version ReconHom(H){\rm Recon_{Hom}}(H) of the HH-colouring problem, in which one must decide for a given GG whether Hom(G,H){\rm Hom}(G,H) is connected, is trivial if and only if HH is dismantlable. We prove a similar starting point for the reconfiguration version of the HH-extension problem. Where Hom(G,H;p){\rm Hom}(G,H;p) is the subgraph of the Hom-graph Hom(G,H){\rm Hom}(G,H) induced by the HH-colourings extending the HH-precolouring pp of GG, the reconfiguration version ReconExt(H){\rm Recon_{Ext}(H)} of the HH-extension problem asks, for a given HH-precolouring pp of a graph GG, if Hom(G,H;p){\rm Hom}(G,H;p) is connected. We show that the graphs HH for which Hom(G,H;p){\rm Hom}(G,H;p) is connected for every choice of (G,p)(G,p) are exactly the NU{\rm NU} graphs. This gives a new characterisation of NU{\rm NU} graphs, a nice class of graphs that is important in the algebraic approach to the CSP{\rm CSP}-dichotomy. We further give bounds on the diameter of Hom(G,H;p){\rm Hom}(G,H;p) for NU{\rm NU} graphs HH, and show that shortest path between two vertices of Hom(G,H;p){\rm Hom}(G,H;p) can be found in parameterised polynomial time. We apply our results to the problem of shortest path reconfiguration, significantly extending recent results.Comment: 17 pages, 1 figur

    Extending precolourings of circular cliques

    Get PDF
    Let G be a graph with circular chromatic number Xe(G) = k/q. Given P ⊆ V (G) where the components of G [P] are isomorphic to the circular clique Gk,g suppose the vertices of P have been precoloured with a (k’,g’) - colouring. We examine under what conditions one can be assured the colouring extends to the entire graph. We stud y sufficient conditions based on k’/q’ − k/q as well as the distance between precoloured components of G[P]. In particular, we examine a conjecture of Albertson and West showing the conditions for extendibility are more complex than anticipated in their work

    Extension from precoloured sets of edges

    Get PDF
    We consider precolouring extension problems for proper edge-colourings of graphs and multigraphs, in an attempt to prove stronger versions of Vizing's and Shannon's bounds on the chromatic index of (multi)graphs in terms of their maximum degree Δ. We are especially interested in the following question: when is it possible to extend a precoloured matching to a colouring of all edges of a (multi)graph? This question turns out to be related to the notorious List Colouring Conjecture and other classic notions of choosability

    Lower Bounds on the Complexity of MSO1 Model-Checking

    Full text link
    One of the most important algorithmic meta-theorems is a famous result by Courcelle, which states that any graph problem definable in monadic second-order logic with edge-set quantifications (i.e., MSO2 model-checking) is decidable in linear time on any class of graphs of bounded tree-width. Recently, Kreutzer and Tazari proved a corresponding complexity lower-bound - that MSO2 model-checking is not even in XP wrt. the formula size as parameter for graph classes that are subgraph-closed and whose tree-width is poly-logarithmically unbounded. Of course, this is not an unconditional result but holds modulo a certain complexity-theoretic assumption, namely, the Exponential Time Hypothesis (ETH). In this paper we present a closely related result. We show that even MSO1 model-checking with a fixed set of vertex labels, but without edge-set quantifications, is not in XP wrt. the formula size as parameter for graph classes which are subgraph-closed and whose tree-width is poly-logarithmically unbounded unless the non-uniform ETH fails. In comparison to Kreutzer and Tazari; (1)(1) we use a stronger prerequisite, namely non-uniform instead of uniform ETH, to avoid the effectiveness assumption and the construction of certain obstructions used in their proofs; and (2)(2) we assume a different set of problems to be efficiently decidable, namely MSO1-definable properties on vertex labeled graphs instead of MSO2-definable properties on unlabeled graphs. Our result has an interesting consequence in the realm of digraph width measures: Strengthening the recent result, we show that no subdigraph-monotone measure can be "algorithmically useful", unless it is within a poly-logarithmic factor of undirected tree-width

    Complexity of colouring problems restricted to unichord-free and \{square,unichord\}-free graphs

    Full text link
    A \emph{unichord} in a graph is an edge that is the unique chord of a cycle. A \emph{square} is an induced cycle on four vertices. A graph is \emph{unichord-free} if none of its edges is a unichord. We give a slight restatement of a known structure theorem for unichord-free graphs and use it to show that, with the only exception of the complete graph K4K_4, every square-free, unichord-free graph of maximum degree~3 can be total-coloured with four colours. Our proof can be turned into a polynomial time algorithm that actually outputs the colouring. This settles the class of square-free, unichord-free graphs as a class for which edge-colouring is NP-complete but total-colouring is polynomial

    Graph Partitioning With Input Restrictions

    Get PDF
    In this thesis we study the computational complexity of a number of graph partitioning problems under a variety of input restrictions. Predominantly, we research problems related to Colouring in the case where the input is limited to hereditary graph classes, graphs of bounded diameter or some combination of the two. In Chapter 2 we demonstrate the dramatic eect that restricting our input to hereditary graph classes can have on the complexity of a decision problem. To do this, we show extreme jumps in the complexity of three problems related to graph colouring between the class of all graphs and every other hereditary graph class. We then consider the problems Colouring and k-Colouring for Hfree graphs of bounded diameter in Chapter 3. A graph class is said to be H-free for some graph H if it contains no induced subgraph isomorphic to H. Similarly, G is said to be H-free for some set of graphs H, if it does not contain any graph in H as an induced subgraph. Here, the set H consists usually of a single cycle or tree but may also contain a number of cycles, for example we give results for graphs of bounded diameter and girth. Chapter 4 is dedicated to three variants of the Colouring problem, Acyclic Colouring, Star Colouring, and Injective Colouring. We give complete or almost complete dichotomies for each of these decision problems restricted to H-free graphs. In Chapter 5 we study these problems, along with three further variants of 3-Colouring, Independent Odd Cycle Transversal, Independent Feedback Vertex Set and Near-Bipartiteness, for H-free graphs of bounded diameter. Finally, Chapter 6 deals with a dierent variety of problems. We study the problems Disjoint Paths and Disjoint Connected Subgraphs for H-free graphs

    Mixing graph colourings

    Get PDF
    This thesis investigates some problems related to graph colouring, or, more precisely, graph re-colouring. Informally, the basic question addressed can be phrased as follows. Suppose one is given a graph G whose vertices can be properly k-coloured, for some k ≥ 2. Is it possible to transform any k-colouring of G into any other by recolouring vertices of G one at a time, making sure a proper k-colouring of G is always maintained? If the answer is in the affirmative, G is said to be k-mixing. The related problem of deciding whether, given two k-colourings of G, it is possible to transform one into the other by recolouring vertices one at a time, always maintaining a proper k-colouring of G, is also considered. These questions can be considered as having a bearing on certain mathematical and ‘real-world’ problems. In particular, being able to recolour any colouring of a given graph to any other colouring is a necessary pre-requisite for the method of sampling colourings known as Glauber dynamics. The results presented in this thesis may also find application in the context of frequency reassignment: given that the problem of assigning radio frequencies in a wireless communications network is often modelled as a graph colouring problem, the task of re-assigning frequencies in such a network can be thought of as a graph recolouring problem. Throughout the thesis, the emphasis is on the algorithmic aspects and the computational complexity of the questions described above. In other words, how easily, in terms of computational resources used, can they be answered? Strong results are obtained for the k = 3 case of the first question, where a characterisation theorem for 3-mixing graphs is given. For the second question, a dichotomy theorem for the complexity of the problem is proved: the problem is solvable in polynomial time for k ≤ 3 and PSPACE-complete for k ≥ 4. In addition, the possible length of a shortest sequence of recolourings between two colourings is investigated, and an interesting connection between the tractability of the problem and its underlying structure is established. Some variants of the above problems are also explored
    corecore