ON INDEPENDENT DOMINATION IN PLANAR 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, i(G), of G is the minimum cardinality of an independent dominating set. Goddard and Henning [Discrete Math. 313 (2013) 839-854] posed the conjecture that if G is not an element of {K-3,K-3, C-5 square K-2} is a connected, cubic graph on n vertices, then i(G) <= 3/8n, where C-5 square K-2 is the 5-prism. As an application of known result, we observe that this conjecture is true when G is 2-connected and planar, and we provide an infinite family of such graphs that achieve the bound. We conjecture that if G is a bipartite, planar, cubic graph of order n, then i(G) <= 1/3n, and we provide an infinite family of such graphs that achieve this bound.