Minimal Dominating Set Enumeration

Years and Authors of Summarized Original Work

• 2011–2014; Kanté, Limouzy, Mary, Nourine, Uno

Problem Definition

Let G be a graph on n vertices and m edges. An edge is written xy (equivalently yx). A dominating set in G is a set of vertices D such that every vertex of G is either in D or is adjacent to some vertex of D. It is said to be minimal if it does not contain any other dominating set as a proper subset. For every vertex x, let N[x] be $\{x\} \cup \{ y\vert xy \in E\}$ and for every $S \subseteq V$ let $N[S] :=\bigcup _{x\in S}N[x]$. For $S \subseteq V$ and x ∈ S we call any $y \in N[x]\setminus N[S\setminus x]$, a private neighbor of x with respect to S. The set of minimal dominating sets of G is denoted by $\mathcal{D}(G)$. We are interested in an output-polynomial algorithm for enumerating $\mathcal{D}(G)$, i.e., listing, without repetitions, all the elements of $\mathcal{D}(G)$ in time bounded by $p\left (n + m,\sum _{D\in \mathcal{D}(G)}\vert D\vert \right )$...