Perfect matching dominating cycle — a new dominating cycle

Abstract

The need for dominating cycles is continuously growing, with the intensification of applied sciences, due to their immense applications especially in networks of all kind. In this paper, new concepts like perfect matching sequences, perfect matching dominating cycles and perfect matching minor are introduced for undirected, loopless, connected even order graphs with a perfect matching. A non repeated finite alternating sequence of edges from a perfect matching and its complement such that each edge is adjacent with the edge preceding and following it, starting and ending at the same edge of the perfect matching is called perfect matching sequence. The spanning subgraph with the edges of such a sequence of length equal to the number of vertices contains a cycle called the perfect matching dominating cycle. The graph obtained by contracting all the edges of a perfect matching is called a perfect matching minor. The existence of a Hamilton cycle in a perfect matching minor of a graph is proved to be a necessary and sufficient condition for the graph to have a perfect matching dominating cycle. These cycles induce new domination parameters. Employing the concept of perfect matching minor, a method to find the number of distinct spanning trees containing a given perfect matching is also briefed.