A generalization of a result of Häggkvist and Nicoghossian

We consider only finite undirected graphs without loops or multiple edges. Our terminology is standard except as indicated. A good reference for any undefined terms is [4]. We begin by introducing some definitions and convenient notation. The set of vertices of a graph G is denoted by V(G) or just V; the set of edges by E(G) or just E. We use K for the connectivity of a graph, and c1 to denote the cardinality of a maximum set of independent vertices in a graph. If C is a cycle in a graph G we call C a dominating cycle if every edge of G has at least one of its vertices on C. Given an (x, y)-path P of G we denote by P the path P with an orientation from x to y. If u, v E V(P), then UPU denotes the consecutive vertices on P from u to u in the direction specified by P. If u # y, then u+ denotes the