Even Cycles And Perfect Matchings In Chordal Graphs

A graph G is called cycle-nice graph if for any even cycle C of G,G-V(C)has a perfect matching.A graph G is called cycle-forced graph if for any even cycle C of G,G-V(C)has a unique perfect matching.A graph G is called PM-compact graph if for any even cycle C of G,G-V(C)has at most one perfect matching.It follows from the definitions that if a graph is cycle-forced,then it is PM-compact and also is cycle-nice.In a graph G,a chord of a cycle C is an edge in E(G)\E(C)such that both of its ends lie in C.A chordal graph is a simple graph in which every cycle of length greater than three has a chord.Let D be an orientation of an undirected graph G.Let C be an even cycle of G,let C be the corresponding directed even cycle of C in D.The cycle C is called oddly-oriented if the number of forward arcs of C is odd.The directed graph D is called Pfaffian if each nice cycle of G is odd-oriented.The graph G is called a Pfaffian graph if G admits a Pfaffian orientation.In this paper,we study cycle-nice chordal graphs and PM-compact chordal graphs.The main results are the following:·A complete characterization of cycle-nice chordal graphs.·Complete characterizations of cycle-forced chordal graphs and chordal graphs with a unique perfect matching.These are also a partial characterization of PM-compact chordal graphs.·Determining the number of perfect matchings of these graphs,characterizing Pfaffian graphs in these classes of graphs.