On (1,2)-domination in cubic graphs

Abstract

A (1,2)-dominating set in a graph G with minimum degree at least 2 is a set S of vertices of G such that every vertex in V(G)∖S has at least one neighbor in S and every vertex in S has at least two neighbors in S. The (1,2)-domination number, γ1,2(G), of G is the minimum cardinality of a (1,2)-dominating set of G. In this paper we prove that if G is a cubic graph of order n, then γ1,2(G)≤34n, and this bound is tight.