Efficient Edge Domination on Hole-Free Graphs in Polynomial Time

Brandstädt, Andreas; Hundt, Christian; Nevries, Ragnar

doi:10.1007/978-3-642-12200-2_56

Cited by 38 publications

(65 citation statements)

References 13 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.…”

mentioning

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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mentioning

confidence: 99%

Efficient domination through eigenvalues

Cardoso

Luz

Pacheco

2016

Discrete Applied Mathematics

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“…To prove it, one has to show that the problem becomes polynomial-time solvable by forbidding any single graph from . However, so far, the conjecture has only been verified for a few forbidden graphs that belong to , and only two of these classes are maximal: 1,2,3 -free graphs 1 [11] and 8 -free graphs [3] (note that 8 = 0, 3,4 ). In the present article, we extend this short list of positive results by one more class where the problem can be solved in polynomial time, namely, the class of 2,2,2 -free graphs.…”

Section: Theorem 11 ( [6] See Section 4) Letmentioning

confidence: 99%

“…(r) Suppose is butterfly-free and contains a vertex with four neighbors 1 , 2 , 3 , 4 such that only two of them are adjacent, say 1 and 2 . If does not contain two vertices 1…”

Section: (D)mentioning

confidence: 99%

“…Since is 2,2,2 -free, we may assume without loss of generality that is the vertex of degree 3 in , and either 1 or 2 , say 1 is a neighbor of in . In other words, ′ contains three vertices 1 , 2 , 3 such that , , , 1 are black, and we obtain a -completion by assigning color black to , and color white to . If 5 is white, then one of 1 , 2 , 3 , 4 is black, say 1 , which means that 2 , 3 , 4 , are black, 1 , are white, and we obtain a -completion by assigning color black to , and color white to .…”

Section: Graph Transformationsmentioning

confidence: 99%

“…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).…”

Section: Introductionmentioning

confidence: 99%

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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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Hereditary Efficiently Dominatable Graphs

Milanič

2012

Journal of Graph Theory

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An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the graph's vertex set. We introduce graphs that are hereditary efficiently dominatable in that sense that every induced subgraph of the graph contains an efficient dominating set. We prove a decomposition theorem for (bull, fork, C 4 )-free graphs, based on which we characterize, in terms of forbidden induced subgraphs, the class of hereditary efficiently dominatable graphs. We also give a decomposition theorem for hereditary efficiently dominatable graphs and examine some algorithmic aspects of such graphs. In particular, we give a polynomial time algorithm for finding an efficient dominating set (if one exists) in a class of graphs properly containing the class of hereditary efficiently dominatable graphs by reducing the problem to the maximum weight independent set problem in claw-free graphs. C 2012 Wiley Periodicals, Inc. Claim 6. If k ≥ 5 then P has no end neighbors.Proof. Let k ≥ 5, and suppose for a contradiction that there exists a vertexSince G has no duplicated vertices, we may assume that there exists a neighbor of v, say w, that is not adjacent to v 1 . Let P = (v, v 2 , . . . , v k ). Then P is a longest induced path of G, and we can apply Claims 4 and 5 to w and P to conclude that N(w) ∩ V (P ) = {v, v 2 }. However, this implies thatcontrary to Claim 4. Claim 7. If k ≥ 5, no vertex dominates P. Journal of Graph Theory Proof. Otherwise, G would contain a semiraft with parts {v 2 , v + 123 } and {v − 234 , v + 34 }. By symmetry, we also obtain: Claim 22. At least one of the sets V + 234 , V − 123 , V + 12 is empty. Claim 23. At least one of the sets V 1234 , V + 123 , V + 34 is empty. Proof. Otherwise, G would contain a semiraft with parts {v + 123 , v 1234 } and {v 4 , v + 34 }. Claim 24. V + 123 = ∅ if and only if V + 34 = ∅. Proof. If V + 123 = ∅ and V + 34 = ∅, then N[v + 34 ] = N[v 4 ], contrary to the fact that G has no duplicated vertices. Conversely, if V + 34 = ∅ and V + 123 = ∅, then N[v + 123 ] = N[v 2 ], which is again a contradiction. HEREDITARY EFFICIENTLY DOMINATABLE GRAPHS 413 By symmetry, we also obtain: Claim 25. V + 234 = ∅ if and only if V + 12 = ∅. Claim 26. V − 123 = ∅ if and only if V − 234 = ∅. Proof. If V − 123 = ∅ and V − 234 = ∅, then N[v − 234 ] = N[v 3 ], contrary to the fact that G has no duplicated vertices. Conversely, if V − 234 = ∅ and V − 123 = ∅, then N[v − 123 ] = N[v 2 ], which is again a contradiction. Claim 27. At least one of the sets V 1234 and V − 234 is empty. Proof. If both sets V 1234 and V − 234 were nonemtpy then G would contain a semiraft with parts {v 2 , v 1234 } ∪ V + 123 and {v − 234 , v 4 } ∪ V 34 + (independently of whether the sets V + 123 and V + 34 are both empty or both nonempty). By symmetry, we also obtain: Claim 28. At least one of the sets V 1234 and V − 123 is empty. Claim 29. If V 1234 = ∅ then V − 34 = ∅. Proof. If V 1234 = ∅ and V − 34 = ∅ then N[v 4 ] = H[v − 34 ], a contradiction with the fact that G has no dupl...

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Efficient Edge Domination on Hole-Free Graphs in Polynomial Time

Cited by 38 publications

References 13 publications

Efficient domination through eigenvalues

Efficient domination through eigenvalues

Dominating induced matchings in graphs containing no long claw

Hereditary Efficiently Dominatable Graphs

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