Hamilton Paths in Dominating Graphs of Trees and Cycles

Abstract

The dominating graph of a graph G has as its vertices all dominating sets of G, with an edge between two dominating sets if one can be obtained from the other by the addition or deletion of a single vertex of G. In this paper we prove that the dominating graph of any tree has a Hamilton path. We also prove that the dominating graph of a cycle on n vertices has a Hamilton path if and only if n ≡ 0 (mod 4).