New Polynomial Cases of the Weighted Efficient Domination Problem

Let G be a finite undirected graph. A vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be NP-complete even for very restricted graph classes. In particular, the ED problem remains NP-complete for 2P3-free graphs and thus for P7-free graphs. We show that the weighted version of the problem (abbreviated WED) is solvable in polynomial time on various subclasses of 2P3-free and P7-free graphs, including (P2+P4)-free graphs, P5-free graphs and other classes. Furthermore, we show that a minimum weight e.d. consisting only of vertices of degree at most 2 (if one exists) can be found in polynomial time. This contrasts with our NP-completeness result for the ED problem on planar bipartite graphs with maximum degree 3

Efficient domination through eigenvalues

The paper begins with a new characterization of (k, τ )-regular sets. Then, using this result as well as the theory of star complements, we derive a simplex-like algorithm for determining whether or not a graph contains a (0, τ )-regular set. When τ = 1, this algorithm can be applied to solve the efficient dominating set problem which is known to be NPcomplete. If −1 is not an eigenvalue of the adjacency matrix of the graph, this particular algorithm runs in polynomial time. However, although it doesn’t work in polynomial time in general, we report on its successful application to a vast set of randomly generated graphs

Investigations in the semi-strong product of graphs and bootstrap percolation

The semi-strong product of graphs G and H is a way of forming a new graph from the graphs G and H. The vertex set of the semi-strong product is the Cartesian product of the vertex sets of G and H, V(G) x V(H). The edges of the semi-strong product are determined as follows: (g1,h1)(g2,h2) is an edge of the product whenever g1g2 is an edge of G and h1h2 is an edge of H or g1 = g2 and h1h2 is an edge of H. A natural subject for investigation is to determine properties of the semi-strong product in terms of those properties of its factors. We investigate distance, independence, matching, and domination in the semi-strong product Bootstrap Percolation is a process defined on a graph. We begin with an initial set of infected vertices. In each subsequent round, uninfected vertices become infected if they are adjacent to at least r infected vertices. Once infected, vertices remain infected. The parameter r is called the percolation threshold. When G is finite, the infection either stops at a proper subset of G or all of V(G) becomes infected. If all of V(G) eventually becomes infected, then we say that the infection percolates and we call the initial set of infected vertices a percolating set. The cardinality of a minimum percolating set of G with percolation threshold r is denoted m(G,r). We determine m(G,r) for certain Kneser graphs and bipartite Kneser graphs

Boundary properties of graphs

A set of graphs may acquire various desirable properties, if we apply suitable restrictions on the set. We investigate the following two questions: How far, exactly, must one restrict the structure of a graph to obtain a certain interesting property? What kind of tools are helpful to classify sets of graphs into those which satisfy a property and those that do not? Equipped with a containment relation, a graph class is a special example of a partially ordered set. We introduce the notion of a boundary ideal as a generalisation of a notion introduced by Alekseev in 2003, to provide a tool to indicate whether a partially ordered set satisfies a desirable property or not. This tool can give a complete characterisation of lower ideals defined by a finite forbidden set, into those that satisfy the given property and to those that do not. In the case of graphs, a lower ideal with respect to the induced subgraph relation is known as a hereditary graph class. We study three interrelated types of properties for hereditary graph classes: the existence of an efficient solution to an algorithmic graph problem, the boundedness of the graph parameter known as clique-width, and well-quasi-orderability by the induced subgraph relation. It was shown by Courcelle, Makowsky and Rotics in 2000 that, for a graph class, boundedness of clique-width immediately implies an efficient solution to a wide range of algorithmic problems. This serves as one of the motivations to study clique-width. As for well-quasiorderability, we conjecture that every hereditary graph class that is well-quasi-ordered by the induced subgraph relation also has bounded clique-width. We discover the first boundary classes for several algorithmic graph problems, including the Hamiltonian cycle problem. We also give polynomial-time algorithms for the dominating induced matching problem, for some restricted graph classes. After discussing the special importance of bipartite graphs in the study of clique-width, we describe a general framework for constructing bipartite graphs of large clique-width. As a consequence, we find a new minimal class of unbounded clique-width. We prove numerous positive and negative results regarding the well-quasi-orderability of classes of bipartite graphs. This completes a characterisation of the well-quasi-orderability of all classes of bipartite graphs defined by one forbidden induced bipartite subgraph. We also make considerable progress in characterising general graph classes defined by two forbidden induced subgraphs, reducing the task to a small finite number of open cases. Finally, we show that, in general, for hereditary graph classes defined by a forbidden set of bounded finite size, a similar reduction is not usually possible, but the number of boundary classes to determine well-quasi-orderability is nevertheless finite. Our results, together with the notion of boundary ideals, are also relevant for the study of other partially ordered sets in mathematics, such as permutations ordered by the pattern containment relation.EThOS - Electronic Theses Online ServiceEngineering and Physical Sciences Research Council (EPSRC)University of Warwick. Centre for Discrete Mathematics and its Applications (DIMAP)GBUnited Kingdo

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

Perfect edge domination : hard and solvable cases

Let G be an undirected graph. An edge of Gdominates itself and all edges adjacent to it. A subset E′ of edges of G is an edge dominating set of G, if every edge of the graph is dominated by some edge of E′. We say that E′ is a perfect edge dominating set of G, if every edge not in E′ is dominated by exactly one edge of E′. The perfect edge dominating problem is to determine a least cardinality perfect edge dominating set of G. For this problem, we describe two NP-completeness proofs, for the classes of claw-free graphs of degree at most 3, and for bounded degree graphs, of maximum degree at most d≥ 3 and large girth. In contrast, we prove that the problem admits an O(n) time solution, for cubic claw-free graphs. In addition, we prove a complexity dichotomy theorem for the perfect edge domination problem, based on the results described in the paper. Finally, we describe a linear time algorithm for finding a minimum weight perfect edge dominating set of a P5-free graph. The algorithm is robust, in the sense that, given an arbitrary graph G, either it computes a minimum weight perfect edge dominating set of G, or it exhibits an induced subgraph of G, isomorphic to a P5.Fil: Lin, Min Chih. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; ArgentinaFil: Lozin, Vadim. University of Warwick; Reino UnidoFil: Moyano, Verónica Andrea. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; ArgentinaFil: Szwarcfiter, Jayme L.. Universidade Federal do Rio de Janeiro; Brasil. Instituto Nacional de Metrologia, Qualidade e Tecnologia; Brasi