New approach to the k-independence number of a graph

Let G = (V,E) be a graph and k > 0 an integer. A k-independent set S V is a set of vertices such that the maximum degree in the graph induced by S is at most k. With k(G) we denote the maximum cardinality of a k-independent set of G. We prove that, for a graph G on n vertices and average degree d, k(G) > k+1 dde+k+1n, improving the hitherto best general lower bound due to Caro and Tuza [Improved lower bounds on k-independence, J. Graph Theory 15 (1991), 99–107].Peer ReviewedPostprint (published version

Degrees in oriented hypergraphs and Ramsey p-chromatic number

The family D(k,m) of graphs having an orientation such that for every vertex v ∈ V (G) either (outdegree) deg+(v) ≤ k or (indegree) deg−(v) ≤ m have been investigated recently in several papers because of the role D(k,m) plays in the efforts to estimate the maximum directed cut in digraphs and the minimum cover of digraphs by directed cuts. Results concerning the chromatic number of graphs in the family D(k,m) have been obtained via the notion of d-degeneracy of graphs. In this paper we consider a far reaching generalization of the family D(k,m), in a complementary form, into the context of r-uniform hypergraphs, using a generalization of Hakimi’s theorem to r-uniform hypergraphs and by showingPeer ReviewedPostprint (published version

Locating-dominating sets and identifying codes in Graphs of Girth at least 5

Locating-dominating sets and identifying codes are two closely related notions in the area of separating systems. Roughly speaking, they consist in a dominating set of a graph such that every vertex is uniquely identified by its neighbourhood within the dominating set. In this paper, we study the size of a smallest locating-dominating set or identifying code for graphs of girth at least 5 and of given minimum degree. We use the technique of vertex-disjoint paths to provide upper bounds on the minimum size of such sets, and construct graphs who come close to meeting these bounds.Award-winningPostprint (author’s final draft

Multiple domination in graphs

Given an undirected and simple graph G = (V , E), a subset D of the vertex set is called a k-dominating set if every vertex not in D has at least k neighbors in D. This concept was introduced by Fink and Jacobson in the year 1985, generalizing the already much studied concept of domination in graphs. In particular, we are interested in finding k-dominating sets of minimum cardinality. Inspired by Fink and Jacobson, Cockayne, Gamble and Shepherd proved in the same year that the k- domination of every graph with minimum degree at least k is at most k/(k + 1) times its order. Since then, the concept of k-domination has gained increased popularity among graph theorists. This thesis aims basically to make a contribution to the study of k-domination in graphs. In the first chapter, we introduce the concepts of domination and k- domination. As it was shown in 1989 by Jacobson and Peters, the problem of finding a minimum k-dominating set belongs to the class of NP-hard problems. However, for some graph classes this problem turns polynomial. We present here a polynomial algorithm for finding a minimum f-dominating set in a block graph, where f-domination is an even more general concept as k-domination. This algorithm comprises previous known ones for trees or rather block graphs. The second chapter handles with different bounds on the k-domination number. First, we present an Erdös-type argument that is useful in proving different inequalities. In particular, beside some new bounds on the k-domination number, we derive a classical bound on the k-domination number due to Caro and Roditty and another of Hopkins and Staton on the k-dependence number. Moreover, we are able to characterize the graphs achieving equality in the bound of Cockayne, Gamble and Shepherd mentioned above. Further, we use a probabilistic method in order to obtain other upper bounds for the k-domination number. As a consequence of one of these probabilistic approaches, it follows a well-known inequality for the usual domination number given by Arnautov, Lovasz and Payan. The last part of this chapter is devoted to the analysis of the graphs achieving equality a bound concerning the k-domination and domination parameters, given by Fink and Jacobson. Here, we present different interesting properties of the extremal graphs. In particular, we show that such graphs contain many induced cycles of length four. Moreover, we characterize the claw-free graphs, the line graphs and the cactus graphs with equal 2-domination and domination numbers. In Chapter 3, we compare the k-domination number with other graph parameters. Further, we analyze the connections between the 2-domination number and the independent domination number, which denotes the minimum cardinality of an independent dominating set in G, and we obtain similar results to previous given ones concerning usual domination. Finally, we explore the relations between the k-domination number and the matching number, the connected domination number and the total domination number. The fourth and last chapter is devoted to special k-domination parameters, where, apart from being k-dominating, we demand the k-dominating set to fulfill further properties, like for example that the underlying induced subgraph is connected or that not only the vertices outside the dominating set but also the vertices inside should be k-dominated. Regarding the respective parameters for the minimum number of vertices required for a subset of vertices in a graph to be k-dominating and satisfying a determined property, we develop some interesting bounds that often either generalize or improve known ones

Multiple domination in graphs

Given an undirected and simple graph G = (V , E), a subset D of the vertex set is called a k-dominating set if every vertex not in D has at least k neighbors in D. This concept was introduced by Fink and Jacobson in the year 1985, generalizing the already much studied concept of domination in graphs. In particular, we are interested in finding k-dominating sets of minimum cardinality. Inspired by Fink and Jacobson, Cockayne, Gamble and Shepherd proved in the same year that the k- domination of every graph with minimum degree at least k is at most k/(k + 1) times its order. Since then, the concept of k-domination has gained increased popularity among graph theorists. This thesis aims basically to make a contribution to the study of k-domination in graphs. In the first chapter, we introduce the concepts of domination and k- domination. As it was shown in 1989 by Jacobson and Peters, the problem of finding a minimum k-dominating set belongs to the class of NP-hard problems. However, for some graph classes this problem turns polynomial. We present here a polynomial algorithm for finding a minimum f-dominating set in a block graph, where f-domination is an even more general concept as k-domination. This algorithm comprises previous known ones for trees or rather block graphs. The second chapter handles with different bounds on the k-domination number. First, we present an Erdös-type argument that is useful in proving different inequalities. In particular, beside some new bounds on the k-domination number, we derive a classical bound on the k-domination number due to Caro and Roditty and another of Hopkins and Staton on the k-dependence number. Moreover, we are able to characterize the graphs achieving equality in the bound of Cockayne, Gamble and Shepherd mentioned above. Further, we use a probabilistic method in order to obtain other upper bounds for the k-domination number. As a consequence of one of these probabilistic approaches, it follows a well-known inequality for the usual domination number given by Arnautov, Lovasz and Payan. The last part of this chapter is devoted to the analysis of the graphs achieving equality a bound concerning the k-domination and domination parameters, given by Fink and Jacobson. Here, we present different interesting properties of the extremal graphs. In particular, we show that such graphs contain many induced cycles of length four. Moreover, we characterize the claw-free graphs, the line graphs and the cactus graphs with equal 2-domination and domination numbers. In Chapter 3, we compare the k-domination number with other graph parameters. Further, we analyze the connections between the 2-domination number and the independent domination number, which denotes the minimum cardinality of an independent dominating set in G, and we obtain similar results to previous given ones concerning usual domination. Finally, we explore the relations between the k-domination number and the matching number, the connected domination number and the total domination number. The fourth and last chapter is devoted to special k-domination parameters, where, apart from being k-dominating, we demand the k-dominating set to fulfill further properties, like for example that the underlying induced subgraph is connected or that not only the vertices outside the dominating set but also the vertices inside should be k-dominated. Regarding the respective parameters for the minimum number of vertices required for a subset of vertices in a graph to be k-dominating and satisfying a determined property, we develop some interesting bounds that often either generalize or improve known ones

Characterization of block graphs with equal 2-domination number and domination number plus one

Let G be a simple graph, and let p be a positive integer. A subset D ⊆ V(G) is a p-dominating set of the graph G, if every vertex v ∈ V(G)-D is adjacent with at least p vertices of D. The p-domination number γₚ(G) is the minimum cardinality among the p-dominating sets of G. Note that the 1-domination number γ₁(G) is the usual domination number γ(G). If G is a nontrivial connected block graph, then we show that γ₂(G) ≥ γ(G)+1, and we characterize all connected block graphs with γ₂(G) = γ(G)+1. Our results generalize those of Volkmann [12] for trees