On the complexity of the dominating induced matching problem in hereditary classes of graphs

Cardoso, Domingos M.; Korpelainen, Nicholas

doi:10.1016/j.dam.2010.03.011

Cited by 41 publications

(67 citation statements)

References 25 publications

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“…This problem is also NP-complete in general [19] and received considerable attention in the literature under several names, such as efficient edge domination or dominating induced matching (see e.g. [3,5,9,10,12,25,26]). An instance of efficient edge domination can be transformed into an instance of efficient domination by associating to the input graph G its line graph L(G), in which case the edges of G become the vertices of L(G) with two vertices being adjacent in L(G) if and only if the respective edges of G share a vertex.…”

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confidence: 99%

Efficient domination through eigenvalues

Cardoso

Luz

Pacheco

2016

Discrete Applied Mathematics

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The paper begins with a new characterization of (k, τ )-regular sets. Then, using this result as well as the theory of star complements, we derive a simplex-like algorithm for determining whether or not a graph contains a (0, τ )-regular set. When τ = 1, this algorithm can be applied to solve the efficient dominating set problem which is known to be NPcomplete. If −1 is not an eigenvalue of the adjacency matrix of the graph, this particular algorithm runs in polynomial time. However, although it doesn't work in polynomial time in general, we report on its successful application to a vast set of randomly generated graphs.

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confidence: 99%

Efficient domination through eigenvalues

Cardoso

Luz

Pacheco

2016

Discrete Applied Mathematics

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“…In this article, we study the problem that appeared in the literature under various names, such as DOMI-NATING INDUCED MATCHING [2,6,7,10,11,13] or EFFICIENT EDGE DOMINATION [1,5,9,15,16], and has several equivalent formulations. One of them, which is used in this article, asks whether the vertices of a graph can be partitioned into two subsets and so that induces a graph of vertex degree 1 (also known as an induced matching) and induces a graph of vertex degree 0 (i.e., an independent set).…”

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Dominating induced matchings in graphs containing no long claw

Hertz

Ries

Zamaraev

et al. 2017

Journal of Graph Theory

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An induced matching M in a graph G is dominating if every edge not in M shares exactly one vertex with an edge in M. The dominating induced matching problem (also known as efficient edge domination) asks whether a graph G contains a dominating induced matching. This problem is generally NP‐complete, but polynomial‐time solvable for graphs with some special properties. In particular, it is solvable in polynomial time for claw‐free graphs. In the present article, we provide a polynomial‐time algorithm to solve the dominating induced matching problem for graphs containing no long claw, that is, no induced subgraph obtained from the claw by subdividing each of its edges exactly once.

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“…In [10], it was shown that the DIM problem is NP-complete; see also [3,9,13,14]. However, for various graph classes, DIM is solvable in polynomial time.…”

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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].…”

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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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On the complexity of the dominating induced matching problem in hereditary classes of graphs

Cited by 41 publications

References 25 publications

Efficient domination through eigenvalues

Efficient domination through eigenvalues

Dominating induced matchings in graphs containing no long claw

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

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