or adjacent to a vertex of $S$. If a dominating set does not contain another dominating set as a subset, it is called a *minimal dominating set*. Throughout this paper we use the abbreviation *mds* for minimal dominating sets.

While the main stream of the research on dominating sets in graphs focuses on the optimization aspects of the problem, there are several interesting graph classes defined around this concept, for example the class of efficient dominating graphs [@Brandstädt2012], of well-dominated graphs [@finbow1988well], of domination perfect graphs [@JGT3190150202], of upper domination perfect [@GUTIN199895] graphs and of strong domination perfect [@RAUTENBACH2001297] graphs.

Another example is the class of *domishold* graphs, introduced by Benzaken and Hammer in [@Benzaken19781]. These are the graphs for which there are positive weights associated to the vertices of the graph such that a subset $D$ of vertices is dominating if and only if the sum of the weights of the vertices of $D$ exceeds a certain threshold $t$. In other words, the characteristic vectors of the dominating sets are exactly the zero-one solutions of a linear inequality, where the coefficients of the inequality correspond to the weights of the vertices.

This concept motivated Payan to define *equidominating* graphs 