Equitable total domination in graphs

A subset ‫ܦ‬ of a vertex set ܸሺ‫ܩ‬ሻ of a graph ‫ܩ‬ ൌ ሺܸ, ‫ܧ‬ሻ is called an equitable dominating set if for every vertex ‫ݒ‬ ‫א‬ ܸ െ ‫ܦ‬ there exists a vertex ‫ݑ‬ ‫א‬ ‫ܦ‬ such that ‫ݒݑ‬ ‫א‬ ‫ܧ‬ሺ‫ܩ‬ሻ and |݀݁݃ሺ‫ݑ‬ሻ െ ݀݁݃ሺ‫ݒ‬ሻ| 1, where ݀݁݃ሺ‫ݑ‬ሻ and ݀݁݃ሺ‫ݒ‬ሻ are denoted as the degree of a vertex ‫ݑ‬ and ‫ݒ‬ respectively. The equitable domination number of a graph ߛ ሺ‫ܩ‬ሻ of ‫ܩ‬ is the minimum cardinality of an equitable dominating set of ‫.ܩ‬ An equitable dominating set ‫ܦ‬ is said to be an equitable total dominating set if the induced subgraph ‫ۄܦۃ‬ has no isolated vertices. The equitable total domination number ߛ ௧ ሺ‫ܩ‬ሻ of ‫ܩ‬ is the minimum cardinality of an equitable total dominating set of ‫.ܩ‬ In this paper, we initiate a study on new domination parameter equitable total domination number of a graph, characterization is given for equitable total dominating set is minimal and also discussed Northaus-Gaddum type results

Semientire Domination in Graphs

ABSTRACT The vertices and edges of a graph G are called the elements of G. We say that a vertex v dominates an edge e if e ∈‫ܰۃ‬ሾ‫ݒ‬ሿ‫.ۄ‬A set D ⊆ V of G = (V, E) is said to be a semientire dominating set if every vertex in V − D is adjacent to at least one vertex in D and every edge in G is dominated by some vertex in D. The semientire domination number ε s (G) of G is the minimum cardinality taken over all the minimal semientire dominating sets of G. In this paper, exact values of ε s (G)for some standard graphs are obtained. Further, bounds on ε s (G)and Nordhaus-Gaddum type results are established