Generating All Minimal Edge Dominating Sets with Incremental-Polynomial Delay

For an arbitrary undirected simple graph G with m edges, we give an algorithm with running time O(m|L|) to generate the set L of all minimal edge dominating sets of G. For bipartite graphs we obtain a better result; we show that their minimal edge dominating sets can be enumerated in time O(m|L|). In fact our results are stronger; both algorithms generate the next minimal edge dominating set with incremental-polynomial delay O(m|L|) and O(m|L|) respectively, when L is the set of already generated minimal edge dominating sets. Our algorithms are tailored for and solve the equivalent problems of enumerating minimal (vertex) dominating sets of line graphs and line graphs of bipartite graphs, with incremental-polynomial delay, and consequently in output-polynomial time. Enumeration of minimal dominating sets in graphs has very recently been shown to be equivalent to enumeration of minimal transversals in hypergraphs. The question whether the minimal transversals of a hypergraph can be enumerated in output-polynomial time is a fundamental and challenging question in Output-Sensitive Enumeration; it has been open for several decades and has triggered extensive research in the field. To obtain our results, we present a flipping method to generate all minimal dominating sets of a graph. Its basic idea is to apply a flipping operation to a minimal dominating set D∗ to generate minimal dominating sets D such that G[D] contains more edges than G[D∗]. Our flipping operation replaces an isolated vertex of G[D∗] with a neighbor outside of D∗, and updates D∗ accordingly to obtain D. The process starts by generating all maximal independent sets, which are known to be minimal dominating sets. Then the flipping operation is applied to every appropriate generated minimal dominating set. We show that the flipping method for enumeration of minimal dominating sets works successfully for line graphs, resulting in an algorithm with incremental-polynomial delay O(nm|L|) on line graphs and an algorithm with incremental-polynomial delay O(nm|L|) on line graphs of bipartite graphs, where n is the number of vertices and L is the set of already generated minimal dominating sets of the input graph. Finally we show that the flipping method also works for graphs of large girth, resulting in an algorithm with incremental-polynomial delay O(nm|L|) to enumerate the minimal dominating sets of graphs of girth at least 7. All given delay times are also the overall running times of the mentioned algorithms, respectively, when L is the set of all minimal dominating sets of the input graph. ∗This work is supported by the European Research Council, the Research Council of Norway, and the French National Research Agency. †Department of Informatics, University of Bergen, Norway, {petr.golovach, pinar.heggernes, yngve.villanger}@ii.uib.no. ‡LITA, Université de Lorraine Metz, France, [email protected]. 1 ar X iv :1 20 8. 53 45 v2 [ cs .D S] 2 3 O ct 2 01 2