Maximum Induced Forests of Product Graphs

The forest number f (G) of a graph G is max{vertical bar S vertical bar : S subset of V(G) and G[S] contains no cycle}. In this paper, we investigate the forest number of several kinds of products of two graphs G and H, such as cartesian product G square H , direct product G x H and lexicographic product G[H]. The sharp bounds are given for Cartesian product and direct product of two graphs. However, characterizing all graphs attaining these bounds is difficult. Among other things, it is shown that (1) for any connected nontrivial graph G of order n with alpha(G) = 2, and any nontrivial forest H, f (G square H) = alpha(G)f (H) + 1 if and only if G congruent to K-n - e and H is an element of {P-3, P-4), or delta(G) = n - 2 and H congruent to K-2. (2) f (G x K-2) = m + 1 for any graph G of order m with delta(G) >= m/2 + 1. (3) For two nonempty graphs G and H, f (G[H]) = alpha(G) f (H). In addition, a number of related conjectures are proposed.