Dominating induced matchings in graphs containing no long claw

Hertz, Alain; Ries, Bernard; Zamaraev, Viktor; Werra, Dominique de

doi:10.1002/jgt.22182

Cited by 19 publications

(34 citation statements)

References 19 publications

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“…One of the most difficult parts of such proofs is finding good transformations. Actually, one not only wants correct transformations but also simple transformations to simplify the proof and as few as possible to avoid long and repetitive proofs (an example with more than 40 transformations can be seen in [21]). We define the simplicity of a transformation as the number of elements of the graph (vertices and edges) touched by the transformation.…”

Section: Transproofmentioning

confidence: 99%

PHOEG Helps to Obtain Extremal Graphsch:32

Devillez

Hauweele

Mélot

2019

Operations Research Proceedings

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Extremal Graph Theory aims to determine bounds for graph invariants as well as the graphs attaining those bounds.We are currently developping PHOEG, an ecosystem of tools designed to help researchers in Extremal Graph Theory.It uses a big relational database of undirected graphs and works with the convex hull of the graphs as points in the invariants space in order to exactly obtain the extremal graphs and optimal bounds on the invariants for some fixed parameters. The results obtained on the restricted finite class of graphs can later be used to infer conjectures. This database also allows us to make queries on those graphs. Once the conjecture defined, PHOEG goes one step further by helping in the process of designing a proof guided by successive applications of transformations from any graph to an extremal graph. To this aim, we use a second database based on a graph data model. The paper presents ideas and techniques used in PHOEG to assist the study of Extremal Graph Theory.

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Section: Transproofmentioning

confidence: 99%

PHOEG Helps to Obtain Extremal Graphsch:32

Devillez

Hauweele

Mélot

2019

Operations Research Proceedings

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“…More recently, the corresponding weighted problem has been solved in O(n) time, by Lin, Mizrahi and Szwarcfiter [12]. Moreover, Hertz, Lozin, Ries, Zamaraev and de Werra [9] have shown that the efficient edge domination problem can still be solved in polynomial time for graphs containing no long claws. Finally, for bounded degree graphs, Cardoso, Cerdeira, Delorme and Silva [4] have shown that efficient edge domination problem is NP-complete for r-regular graphs for r ≥ 3.…”

Section: Efficient Edge Dominationmentioning

confidence: 99%

“…The problems consist in determining such dominating sets, of minimum cardinality or minimum total weight of their edges. Most of the known results, so far concerned the efficient edge dominating case (also known as dominating induced matching), for instance see [2,3,4,6,8,9,10,11,12,13,14,15,16,17]. These two types of dominations may lead to problems of a quite different nature.…”

Section: Introductionmentioning

confidence: 99%

Perfect edge domination: hard and solvable cases

2017

Self Cite

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Let G be an undirected graph. An edge of G dominates itself and all edges adjacent to it. A subset E ′ of edges of G is an edge dominating set of G, if every edge of the graph is dominated by some edge of E ′ . We say that E ′ is a perfect edge dominating set of G, if every edge not in E ′ is dominated by exactly one edge of E ′ . The perfect edge dominating problem is to determine a least cardinality perfect edge dominating set of G. For this problem, we describe two NP-completeness proofs, for the classes of claw-free graphs of degree at most 3, and for bounded degree graphs, of maximum degree at most d ≥ 3 and large girth. In contrast, we prove that the problem admits an O(n) time solution, for cubic claw-free graphs. In addition, we prove a complexity dichotomy theorem for the perfect edge domination problem, based on the results described in the paper. Finally, we describe a linear time algorithm for finding a minimum weight perfect edge dominating set of a P 5 -free graph. The algorithm is robust, in the sense that, given an arbitrary graph G, either it computes a minimum weight perfect edge dominating set of G, or it exhibits an induced subgraph of G, isomorphic to a P 5 .

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“…The following results are known: Theorem 1. DIM is solvable in polynomial time for (i) S 1,1,1 -free graphs [9], (ii) S 1,2,3 -free graphs [12], (iii) S 2,2,2 -free graphs [11], (iv) S 1,2,4 -free graphs [6], (v) S 2,2,3 -free graphs [7], (vi) S 1,1,5 -free graphs [8], (vii) P 7 -free graphs [4] (in this case even in linear time), (viii) P 8 -free graphs [5].…”

mentioning

confidence: 99%

“…In [11], it is conjectured that for every fixed i, j, k, DIM is solvable in polynomial time for S i,j,k -free graphs (actually, an even stronger conjecture is mentioned in [11]); this includes P k -free graphs for k ≥ 9. In this paper we show that DIM can be solved in polynomial time for P 9 -free graphs (generalizing the corresponding results for P 7 -free and for P 8 -free graphs).…”

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confidence: 99%

Finding dominating induced matchings in S1,1,5-free graphs in polynomial time

Brandstädt

Mosca

2020

Discrete Applied Mathematics

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Let G = (V, E) be a finite undirected graph. An edge subset E ′ ⊆ E is a dominating induced matching (d.i.m.) in G if every edge in E is intersected by exactly one edge of E ′ . The Dominating Induced Matching (DIM ) problem asks for the existence of a d.i.m. in G. The DIM problem is NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree 3 but was solved in linear time for P 7 -free graphs and in polynomial time for P 8 -free graphs. In this paper, we solve it in polynomial time for P 9 -free graphs.Let G = (V, E) be a finite simple undirected graph, i.e., an undirected graph without loops and multiple edges. Given an edge e ∈ E, we say that e dominates itself and every edge sharing a vertex with e. An edge subset M ⊆ E is an induced matching if the pairwise distance between its members is at least 2, that is, M is isomorphic to kP 2 for k = |M |. A subset M ⊆ E is a dominating induced matching (d.i.m. for short) of G if M is an induced matching in G such that every edge in E is dominated by exactly one edge in M . Clearly, not every graph G has a d.i.m.; the Dominating Induced Matching (DIM) problem asks for the existence of a d.i.m. in G.The DIM problem is also called Efficient Edge Domination (EED) in various papers: Recall that a vertex v ∈ V dominates itself and its neighbors. A vertex subset D ⊆ V is an efficient dominating set (e.d.s. for short) of G if every vertex of G is dominated by exactly one vertex in D. The notion of efficient domination was introduced by Biggs [2] under the name perfect code. The Efficient Domination (ED) problem asks for the existence of an e.d.s. in a given graph G (note that not every graph has an e.d.s.) A set M of edges in a graph G is an efficient edge dominating set (e.e.

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Dominating induced matchings in graphs containing no long claw

Cited by 19 publications

References 19 publications

PHOEG Helps to Obtain Extremal Graphsch:32

PHOEG Helps to Obtain Extremal Graphsch:32

Perfect edge domination: hard and solvable cases

Finding dominating induced matchings in S1,1,5-free graphs in polynomial time

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