EQUITABLE DOMINATION IN GRAPHS

Abstract

Let D be a dominating set of a graph G = (V, E). For v ∈ D, let n1(v) = |N(v) ∩ (V - D)| and for w ∈ V - D, let n2(w) = |N(w) ∩ D|. Then D is called an equitable dominating set of type 1 if |n1(v1) - n1(v2)| ≤ 1 for all v1, v2 ∈ D and is called an equitable dominating set of type 2 if |n2(w1) - n2(w2)| ≤ 1 for all w1, w2 ∈ V - D. The minimum cardinality of an equitable dominating set of G of type 1 (type 2) is called the 1-equitable (2-equitable) domination number of G and is denoted by γ eq1 (G)(γ eq2 (G)). If D is an equitable dominating set of type 1 and type 2, then D is called an equitable dominating set and the equitable domination number of G is defined to be the minimum cardinality of an equitable dominating set and is denoted by γ eq (G). In this paper we initiate a study of these parameters.