A set S of vertices in a graph G is dominating if every vertex not adjacent to S. If, addition, an independent set, then set. The domination number i(G) the minimum cardinality G. In Goddard and Henning (Discrete Math 313:839–854, 2013) conjectured that connected cubic order n, $$i(G) \le \frac{3}{8}n$$ , except complete bipartite $$K_{3,3}$$ or 5-prism $$C_5 \, \Box K_2$$ . Further they constr...

Abstract A set S of vertices in a graph G is dominating if every vertex not ad jacent to . If, addition, an independent set, then set. The domination number i ( ) the minimum cardinality subdivision $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G<...

A subset D of ( ) V G is called an equitable dominating set if for every ( ) v V G D   there exists a vertex u D  such that ( ) uv E G  and | ( ) ( ) | 1 deg u deg v   . A subset D of ( ) V G is called an equitable independent set if for any , u D v   ( ) e N u for all { } v D u   . The concept of equi independent equitable domination is a combination of these two important concepts. ...

In this paper, we show that the Italian domination number of every lexicographic product graph G ○ H can be expressed in terms five different parameters . These defined under following unified approach, which encompasses definition several well-known and introduces new ones. Let N ( v ) denote open neighbourhood ∈ V , let w = 0 1 …, l a vector nonnegative integers such ≥ We say function f : → {...

In a graph G = (V (G), E(G)), a vertex dominates itself and its neighbors. A subset S of V (G) is a double dominating set if every vertex of V (G) is dominated at least twice by the vertices of S. The double domination number of G is the minimum cardinality among all double dominating sets of G. We consider the effects of edge removal on the double domination number of a graph. We give a necess...