A Roman dominating function of a graph G is a labeling f : V (G) −→ {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number γR(G) of G is the minimum of ∑ v∈V (G) f(v) over such functions. The Roman domination subdivision number sdγR(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order t...

For a graph G = (V, E), a set S ⊆ V (G) is a total dominating set if it is dominating and both 〈S〉 has no isolated vertices. The cardinality of a minimum total dominating set in G is the total domination number. A set S ⊆ V (G) is a total restrained dominating set if it is total dominating and 〈V (G) − S〉 has no isolated vertices. The cardinality of a minimum total restrained dominating set in ...

Vizing’s conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this paper we survey the approaches to this central conjecture from domination theory and give some new results along the way. For instance, several new properties of a minimal counterexample to the conjecture are obtained an...

The problem of monitoring an electric power system is placing as few measurement devices as possible. In graph theoretical representation, it can be considered as a variant of domination problem, namely, power domination problem. This problem is to find a minimum power domination set S of a graph G = (V,E) with S ⊆ V and S can dominate all vertices and edges through the observation rules accord...

A subset S of V is called a total dominating set if every vertex in V is adjacent to some vertex in S. The total domination number γt (G) of G is the minimum cardinality taken over all total dominating sets of G. A dominating set is called a connected dominating set if the induced subgraph 〈S〉 is connected. The connected domination number γc(G) of G is the minimum cardinality taken over all min...

A set D of vertices in a graph G = (V, E) is a weakly connected dominating set of G if D is dominating in G and the subgraph weakly induced by D is connected. The weakly connected domination number of G is the minimum cardinality of a weakly connected dominating set of G. The weakly connected domination subdivision number of a connected graph G is the minimum number of edges that must be subdiv...

A graph G with no isolated vertex is total domination bicritical if the removal of any pair of vertices, whose removal does not produce an isolated vertex, decreases the total domination number. In this paper we study properties of total domination bicritical graphs, and give several characterizations.

The aim of this paper is to obtain closed formulas for the perfect domination number, Roman number and lexicographic product graphs. We show that these can be obtained relatively easily case first two parameters. picture quite different when it concerns number. In case, we general bounds then give sufficient and/or necessary conditions achieved. also discuss graphs characterize where equals

A subset S of the vertices of a graph G is an outer-connected dominating set, if S is a dominating set of G and G − S is connected. The outer-connected domination number of G, denoted by γ̃c(G), is the minimum cardinality of an OCDS of G. In this paper we generalize the outer-connected domination in graphs. Many of the known results and bounds of outer-connected domination number are immediate c...

A sequence of vertices in a graph G is called a legal dominating sequence if every vertex in the sequence dominates at least one vertex not dominated by those vertices that precede it, and at the end all vertices of G are dominated. While the length of a shortest such sequence is the domination number of G, in this paper we investigate legal dominating sequences of maximum length, which we call...