Reducible chains of planar 1-cycle resonant graphs Original Research Article

A connected graph G is said to be k-cycle resonant if, for 1⩽t⩽k, any t disjoint cycles in G are mutually resonant, that is, there is a perfect matching M of G such that each of the t cycles is an M-alternating cycle. The concept of k-cycle resonant graphs was introduced by the present authors in 1994. Some necessary and sufficient conditions for a graph to be k-cycle resonant were given. Recently, some necessary and sufficient conditions for a planar graph to be 1-cycle resonant and 2-cycle resonant were also given. In this paper, we investigate 1-cycle-resonant reducible (simply 1-CR-reducible) chains of planar 1-cycle resonant graphs. The upper and lower bounds of the numbers of 1-CR-reducible chains of planar 1-cycle resonant graphs are given. Consequently, it is shown that any planar 1-cycle resonant graph has an ear decomposition. Furthermore, a construction method of planar 1-cycle resonant graphs and an efficient algorithm for recognizing 1-cycle resonant graphs are established.