A set S ⊆ V of vertices in a graph G = (V,E) without isolated vertices is a total dominating set if every vertex of V is adjacent to some vertex in S. The total domination number γt(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sdγt(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in...

For a fixed positive integer k, a k-tuple total dominating set of a graph G is a subset D ⊆ V (G) such that every vertex of G is adjacent to at least k vertices in D. The k-tuple total domination problem is to determine a minimum k-tuple total dominating set of G. This paper studies k-tuple total domination from an algorithmic point of view. In particular, we present a linear-time algorithm for...

In this paper we continue the investigation of total domination in Cartesian products of graphs first studied in Graphs Combin. 21 (2005), 63–69. A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to some vertex in S. The maximum cardinality of a minimal total dominating set of G is the upper total domination number of G, denoted by Γt(G). We prove ...

The domination problem and its variants have been extensively studied in the literature. In this paper we investigate the domination problem in distance-hereditary graphs. In particular, we give a linear-time algorithm for the domination problem in distance-hereditary graphs by a labeling approach. We actually solve a more general problem, called the L-domination problem, which also includes th...

Let G = (V,E) be a graph without isolated vertices. A set S ⊆ V is a total dominating set if every vertex in V is adjacent to at least one vertex in S. A total dominating set S ⊆ V is a paired-dominating set if the induced subgraph G[S] has at least one perfect matching. The paired-domination number γpr(G) is the minimum cardinality of a paired-domination set of G. In this paper, we provide a c...

Abstract. In this paper we give tight upper bounds on the total domination number, the weakly connected domination number and the connected domination number of a graph in terms of order and Euler characteristic. We also present upper bounds for the restrained bondage number, the total restrained bondage number and the restricted edge connectivity of graphs in terms of the orientable/nonorienta...

A set S of vertices of a graph G= (V ,E) with no isolated vertex is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. The total domination number t(G) is the minimum cardinality of a total dominating set ofG. The total domination subdivision number sd t (G) is the minimum number of edges that must be subdivided in order to increase the total domination number. We ...