Strong weak domination and domination balance in a graph Original Research Article

Let G = (V, E) be a graph and u, v ∈ V. Then, ustrongly dominates v and v weakly dominates u if (i) uv ∈ E and (ii) deg u ⩾ deg v. A set D ⊂ V is a strong-dominating set (sd-set) of G if every vertex in V − D is strongly dominated by at least one vertex in D. Similarly, a weak-dominating set (wd-set) is defined. The strong (weak) domination number γs (γw) of G is the minimum cardinality of an sd-set (wd-set). Besides investigating some relationship of γs and γw with other known parameters of G, some bounds are obtained. A graph G is domination balanced if there exists an sd-set D1 and a wd-set D2 such that D1 ∩ D2 = 0. A study of domination balanced graphs is initiated.