Neighborhood preferences for minimal dominating sets enumeration
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We investigate two different approaches to enumerate minimal dominating sets of a graph using structural properties based on neighborhood inclusion. In the first approach, we define a preference relation on a graph G as a poset on the set of vertices of G. In particular, we consider the poset of closed neighborhood inclusion P(G) and define the notion of preferred dominating set as dominating sets that correspond to minimal ideals of P(G). We show that graphs with a unique preferred dominating set are those who are dominated by simplicial vertices and show that there is a polynomial delay algorithm to enumerate minimal dominating sets if there is one to enumerate preferred dominating sets. In the second approach we consider intersections of minimal dominating sets with redundant vertices, i.e., vertices that are not minimal in P(G). We show in a generalized class of split graphs that there is a linear delay algorithm to enumerate minimal dominating sets if these intersections form an independent set system. Graphs that share this property include completed P7-free chordal graphs which improves results from [14] on P6-free chordal graphs.
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