On graphs whose domination numbers equal their independent domination numbers

Abstract

Abstract In this paper, we extend a result due to R. B. Allan and R. C. Laskar on graphs whose independent domination numbers equal their domination numbers. We will consider finite simple graphs as treated in most of the standard text-books on Graph Theory (e.g., see D. B. West [1]). Let G = (V,E) be any graph and D ⊆ V. We let N(D) denote the set of all vertices adjacent to those in D; the vertices in N(D) are often called the neighbours of those in D. In particular, if D = {u} then N(D) is written simply as N(u) The set D is called a dominating set (or, simply a domset) of G if N(ν) ∩ D ≠ ∅ for every ν ∈ V - D and an independent domset is often called a kernel of G (see [2, 3, 4, 5]). We will denote by D(G) (Di(G)) the set of all domsets (kernels) in G If G is a finite graph then the least cardinality of a domset (kernel) of G is called its domination (independent domination) number, denoted γ(G) (γi(G)). The following result is well known. We thank the referees for pointing out an error in the original version of the paper which led to substantial improvement in the main result of this note.