A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The total domination number γt(G) is the minimum cardinality of a total dominating set of G. A vertex that is contained in some minimum total dominating set of a graph G is a good vertex, otherwise it is a bad vertex. We determine for which triples (x, y, z) there exists a conne...

Domination in graphs has been an extensively researched branch of graph theory. Graph theory is one of the most flourishing branches of modern mathematics and computer applications. An introduction and an extensive overview on domination in graphs and related topics is surveyed and detailed in the two books by Haynes et al. [ 1, 2]. Recently dominating functions in domination theory have receiv...

The bondage number of a graph G is the minimum number of edges whose removal results in a graph with larger domination number.A dominating setD is called an efficient dominating set ofG if |N−[v]∩D|=1 for every vertex v ∈ V (G). In this paper we establish a tight lower bound for the bondage number of a vertex-transitive graph. We also obtain upper bounds for regular graphs by investigating the ...

Given a vertex-coloured graph, a dominating set is said to be tropical if every colour of the graph appears at least once in the set. Here, we study minimum tropical dominating sets from structural and algorithmic points of view. First, we prove that the tropical dominating set problem is NP-complete even when restricted to a simple path. Then, we establish upper bounds related to various param...

An upper dominating set in a graph is a minimal (with respect to set inclusion) dominating set of maximum cardinality. The problem of finding an upper dominating set is generally NP-hard, but can be solved in polynomial time in some restricted graph classes, such as P4-free graphs or 2K2-free graphs. For classes defined by finitely many forbidden induced subgraphs, the boundary separating diffi...

A vertex subset S of a graph G is a perfect (resp. quasiperfect) dominating set in G if each vertex v of G \ S is adjacent to only one vertex (dv ∈ {1, 2} vertices) of S. Perfect and quasiperfect dominating sets in the regular tessellation graph of Schläfli symbol {3, 6} and in its toroidal quotients are investigated, yielding the classification of their perfect dominating sets and most of thei...

The class of graphs that do not contain an induced path on k vertices, Pk-free graphs, plays a prominent role in algorithmic graph theory. This motivates the search for special structural properties of Pkfree graphs, including alternative characterizations. Let G be a connected Pk-free graph, k ≥ 4. We show that G admits a connected dominating set that induces either a Pk−2-free graph or a grap...

We study dominating sets whose induced subgraphs have a bounded diameter. Such sets were used in recent papers by Kim et al. and Yu et al. to model virtual backbones in wireless networks where the number of hops required to forward messages within the backbone is minimized. We prove that for any fixed k ≥ 1 it is NP-complete to decide whether a given graph admits a dominating set whose induced ...

Let G be an undirected graph. An edge of G dominates 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 perf...

Bidimensionality is a powerful tool for developing subexponential fixed-parameter algorithms for combinatorial optimization problems on graph families that exclude a minor. This paper completes the theory of bidimensionality for graphs of bounded genus (which is a minorexcluding family). Specifically we show that, for any problem whose solution value does not increase under contractions and who...