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Enumerating Minimal Dominating Sets in Triangle-Free Graphs

Marthe Bonamy, Oscar Defrain, Marc Heinrich, Jean-Florent Raymond, Michael Wagner

2019
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Symposium on Theoretical Aspects of Computer Science
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It is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this paper we prove that this is the case in triangle-free graphs. This answers a question of Kanté et al. Additionally, we show that deciding if a set of vertices of a bipartite graph can be completed into a minimal dominating set is a NP-complete problem. ACM Subject Classification Mathematics of computing → Graph algorithms; Theory of computation → Design and
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... sis of algorithms Acknowledgements The authors wish to thank Paul Ouvrard for extensive discussions on the topic of this paper. We also gratefully acknowledge support from Nicolas Bonichon and the Simon family for the organization of the 3 rd Pessac Graph Workshop, where this research was done. Last but not least, we thank Peppie for her unwavering support during the work sessions. The objects we wish to enumerate in this paper are the (inclusion-wise) minimal dominating sets of a given graph. In general, the number of these objects may grow exponentially with the order n of the input graph. Therefore, in stark contrast to decision or optimization problems, looking for a running time polynomially bounded by n is not a reasonable, let alone meaningful, efficiency criterion. Rather, we aim here for algorithms whose running time is polynomially bounded by the size of both the input and output data, called output-polynomial algorithms. Because dominating sets are among the most studied objects in graph theory and algorithms, their enumeration (and counting) have attracted an increasing attention over the past 10 years. The problem of enumerating minimal dominating sets (hereafter referred to as Dom-Enum) has a notable feature: it is equivalent to the extensively studied hypergraph problem Trans-Enum. In Trans-Enum, one is given a hypergraph H (i.e. a collection of sets, called hyperedges) and is asked to enumerate all the minimal transversals of H (i.e. the inclusion-minimal sets of elements that meet every hyperedge). It is not hard to see that Dom-Enum is a particular case of Trans-Enum: the minimal dominating sets of a graph G are exactly the minimal transversals of the hypergraph of closed neighborhoods of G. Conversely, Kanté, Limouzy, Mary, and Nourine proved that every instance of Trans-Enum can be reduced to a co-bipartite 1 instance of Dom-Enum [17] . Currently, the best output-sensitive algorithm for Trans-Enum is due to Fredman and Khachiyan and runs in quasi-polynomial time [9] . It is a long-standing open problem whether this complexity bound can be improved (see for instance the surveys [6, 8] ). Therefore, the equivalence between the two problems is an additional motivation to study Dom-Enum, with the hope that techniques from graph theory will be used to obtain new results on the Trans-Enum problem. So far, output-polynomial algorithms have been obtained for Dom-Enum in several classes of graphs, including planar graphs and degenerate graphs [7], classes of graphs of bounded tree-width, clique-width [4], or mim-width [10], path graphs and line graphs [16], interval graphs and permutation graphs [18], split graphs [19], graphs of girth at least 7 [12], chordal graphs [19], and chordal bipartite graphs [11]. A succinct survey of results on Dom-Enum can be found in [20] . The authors of [19] state as an open problem the question to design an output-polynomial algorithm for bipartite graphs (the problem also appeared in [20, 11] ). We address this problem with the following result. S TA C S 2 0 1 9

doi:10.4230/lipics.stacs.2019.16
dblp:conf/stacs/BonamyDHR19
fatcat:2qvswqgw3fez5l3fs44c4urql4