Independent domination in bipartite 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 talk, we study the following conjecture posed by Goddard and Henning [Discrete Math. 313 (2013), 839-854]: If G 6= K_3,3 is a connected, cubic, bipartite graph on n vertices, then i(G) = ₁₁⁴ n. As a consequence of a more general result due to Dorbec et al. [J. Graph Theory 80(4) (2015), 329-349], it is known that if G is a bipartite, connected, cubic, graph of order n that does not have an induced subgraph isomorphic to a supergraph of K_2,3, then i(G) = ³₈n. In this talk, we improve this ³₈-upper bound to a ₁₁⁴ -upper bound, thereby proving the Goddard-Henning conjecture when the graph does not have an induced subgraph isomorphic to a supergraph of K_2,3. This strengthens a result in [Discrete Applied Math. 162 (2014), 399-403] which proves the Goddard-Henning conjecture when the girth is at least 6.