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 article, we show that if G ≠ C 5□K 2is a connected cubic graph of order  n that does not have a subgraph isomorphic to K 2, 3 , then i ( G ) ≤ 3 n / 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 ) ≤ 3 n / 8 , where γ ( G ) denotes the domination number of G .