2020
DOI: 10.1145/3386686
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Enumerating Minimal Dominating Sets in Kt-free Graphs and Variants

Abstract: It is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this article we investigate this problem in graph classes defined by forbidding an induced subgraph. In particular, we provide output-polynomial time algorithms for K t -free graphs and for several related graph classes. This answers a question of Kanté et al. about enumeration in bipartite graphs.

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Cited by 16 publications
(6 citation statements)
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“…Assume that a search-tree node corresponds to a partial solution (or pre-solution) U and instead of proceeding with the search-tree algorithm (by exploring all the possible paths from this node onward) we ask whether we can extend U to a meaningful solution S. In the case of Dominating Set, this means that S is an inclusion-wise minimal dominating set that contains U . Unfortunately, this Extension Dominating Set problem and many similar problems are NP-hard, see [6,12,14,15,37,38,45]. Even worse: when parameterized by the "pre-solution size," Extension Dominating Set is one of the few problems known to be complete for the parameterized complexity class W [3], as shown in [11].…”
Section: Introductionmentioning
confidence: 99%
“…Assume that a search-tree node corresponds to a partial solution (or pre-solution) U and instead of proceeding with the search-tree algorithm (by exploring all the possible paths from this node onward) we ask whether we can extend U to a meaningful solution S. In the case of Dominating Set, this means that S is an inclusion-wise minimal dominating set that contains U . Unfortunately, this Extension Dominating Set problem and many similar problems are NP-hard, see [6,12,14,15,37,38,45]. Even worse: when parameterized by the "pre-solution size," Extension Dominating Set is one of the few problems known to be complete for the parameterized complexity class W [3], as shown in [11].…”
Section: Introductionmentioning
confidence: 99%
“…This paper combines four lines of research: (a) studying variations of domination problems, here the Roman domination [17,21,28]; (b) input-sensitive enumeration of minimal solutions, a topic that has drawn attention in particular from people also interested in domination problems [2,18,19,26,27]; (c) related to (and motivated by) enumeration, extension problems have been introduced and studied in particular in the context of domination problems 3 in [3,9,11,12,32,33,40]: is a given set a subset of any minimal dominating set? ; (d) the Hitting Set Transversal Problem is the question if all minimal hitting sets of a hypergraph can be enumerated with polynomial delay (or even output-polynomial) only: this question is open for four decades by now and is equivalent to several enumeration problems in logic, database theory and also to enumerating minimal dominating sets in graphs, see [20,22,25,31].…”
Section: Introductionmentioning
confidence: 99%
“…Enumeration of minimal (edge) dominating sets is a central topic of the field of enumeration. Thus, without cardinality constraints, the problem has hardness results and positive results for various graph classes [5,[15][16][17]. For minimal edge dominating set enumeration, Kanté et al have been developed a polynomial-delay and polynomial-space algorithm [18].…”
Section: Introductionmentioning
confidence: 99%