On cycle resonant outerplanar graphs

A connected graph G is said to be k-cycle resonant if, for 0 <= t <= k, for any t disjoint cycles C-1, C-2, ... , C-t in G, there is a perfect matching in G - U-i=1(t) V(C-i). A connected graph G is said to be cycle resonant if G is k*-cycle resonant and k* is the maximum number of disjoint cycles in G. In this paper we prove that for outerplane graphs, 2-cycle resonant is equivalent to cycle resonant and establish a necessary and sufficient condition for an outerplanar graph to be cycle resonant. We also discuss the structure of 2-connected cycle resonant outerplane graphs. Let Phi(G) denote the number of perfect matchings in G. For any 2-connected cycle resonant outerplane graph G with k chords, we get k + 2 <= Phi(G) <= 2(k) + 1 and give the extremal graphs for the equalities in the inequalities.