Independent Domination in Cubic Graphs

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number of G, denoted by i(G), is the minimum cardinality of an independent dominating set. In this paper, we show that if G �= C5 ✷K2 is a connected cubic graph of order n that does not have a subgraph isomorphic to K2,3, then i(G) ≤ 3n/8. As a consequence of our main result, we deduce Reed’s important result [Combin. Probab. Comput. 5 (1996), 277–295] that if G is a cubic graph of order n, then γ(G) ≤ 3n/8, where γ(G) denotes the domination number of G.