Let F1, F2, . . . , Fk be graphs with the same vertex set V . A subset S ⊆ V is a factor dominating set if in every Fi every vertex not in S is adjacent to a vertex in S, and a factor total dominating set if in every Fi every vertex in V is adjacent to a vertex in S. The cardinality of a smallest such set is the factor (total) domination number. In this note we investigate bounds on the factor ...

A secure (total) dominating set of a graph G = (V, E) is a (total) dominating set X ⊆ V with the property that for each u ∈ V − X , there exists x ∈ X adjacent to u such that (X − {x}) ∪ {u} is (total) dominating. The smallest cardinality of a secure (total) dominating set is the secure (total) domination number γs(G) (γst(G)). We characterize graphs with equal total and secure total domination...

I. Introduction In this paper, D=(V, A) is a finite, directed graph with neither loops nor multiple arcs (but pairs of opposite arcs are allowed) and G=(V, E) is a finite, undirected graph with neither loops nor multiple edges. For basic terminology, we refer to Chartrand and Lesniak [2]. A set S of vertices in a graph G=(V, E) is a dominating set if every vertex in V – S is adjacent to some ve...

Recently, Azarija et al. considered the prism G K2 of a graph G and showed that γt(G K2) = 2γ(G) if G is bipartite, where γt(G) and γ(G) are the total domination number and the domination number of G. In this note, we give a simple proof and observe that there are similar results for other pairs of parameters. We also answer a question from that paper and show that for all graphs γt(G K2) ≥ 4 3...

The domination number γ(G) and the total domination number γt(G) of a graph G without an isolated vertex are among the most well studied parameters in graph theory. While the inequality γt(G) ≤ 2γ(G) is an almost immediate consequence of the definition, the extremal graphs for this inequality are not well understood. Furthermore, even very strong additional assumptions do not allow to improve t...

We present the first polynomial time algorithms for solving the NPcomplete graph problems DOMINATING SET and TOTAL DOMINATING SET when restricted to asteroidal triple-free graphs. We also present algorithms to compute a minimum cardinality dominating set and a minimum cardinality total dominating set on a large superclass of the asteroidal triple-free graphs, called DDP-graphs. These algorithms...

Let F1, F2, . . . , Fk be graphs with the same vertex set V . A subset S ⊆ V is a factor dominating set if in every Fi every vertex not in S is adjacent to a vertex in S, and a factor total dominating set if in every Fi every vertex in V is adjacent to a vertex in S. The cardinality of a smallest such set is the factor (total) domination number. We investigate bounds on the factor (total) domin...

A graph G is said to be k-γt-critical if the total domination number γt(G) = k and γt(G + uv) < k for every uv / ∈ E(G). A k-γc-critical graph G is a graph with the connected domination number γc(G) = k and γc(G + uv) < k for every uv / ∈ E(G). Further, a k-tvc graph is a graph with γt(G) = k and γt(G− v) < k for all v ∈ V (G), where v is not a support vertex (i.e. all neighbors of v have degre...

Several of the best known problems and conjectures in graph theory arise in studying the behavior of a graphical invariant on a graph product. Examples of this are Vizing’s conjecture, Hedetniemi’s conjecture and the calculation of the Shannon capacity of graphs, where the invariants are the domination number, the chromatic number and the independence number on the Cartesian, categorical and st...