Graphs such that all minimum dominating sets intersect all maximally independent sets

Abstract

In earlier work, the connected graphs with the property announced in the title and domination number two were completely characterized. Here we describe infinitely many connected graphs of all domination numbers > 1 with the property, prove a structure theorem about such graphs with domination number three, and show that the smallest connected graph with domination number three and this property is two cycles of length four connected by a path of length two.