Acyclic Matching in Some Subclasses of Graphs

Panda, B. S.; Chaudhary, Juhi

doi:10.1007/978-3-030-48966-3_31

Cited by 13 publications

(4 citation statements)

References 10 publications

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“…The problems involving matchings have a vast literature in both structural and algorithmic graph theory [13,21,23,24,25,26,27]. A matching is a subset M ⊆ E of edges of a graph G = (V, E) that do not share any endpoint.…”

Section: Introductionmentioning

confidence: 99%

Weighted Connected Matchings

Gomes¹,

Masquio²,

Pinto³

et al. 2022

Preprint

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A matching M is a P-matching if the subgraph induced by the endpoints of the edges of M satisfies property P. As examples, for appropriate choices of P, the problems Induced Matching, Uniquely Restricted Matching, Connected Matching and Disconnected Matching arise. For many of these problems, finding a maximum P-matching is a knowingly NP-hard problem, with few exceptions, such as connected matchings, which has the same time complexity as the usual Maximum Matching problem. The weighted variant of Maximum Matching has been studied for decades, with many applications, including the well-known Assignment problem. Motivated by this fact, in addition to some recent researches in weighted versions of acyclic and induced matchings, we study the Maximum Weight Connected Matching. In this problem, we want to find a matching M such that the endpoint vertices of its edges induce a connected subgraph and the sum of the edge weights of M is maximum. Unlike the unweighted Connected Matching problem, which is in P for general graphs, we show that Maximum Weight Connected Matching is NP-hard even for bounded diameter bipartite graphs, starlike graphs, planar bipartite, and bounded degree planar graphs, while solvable in linear time for trees and subcubic graphs. When we restrict edge weights to be non negative only, we show that the problem turns to be polynomially solvable for chordal graphs, while it remains NP-hard for most of the cases when weights can be negative. Our final contributions are on parameterized complexity. On the positive side, we present a single exponential time algorithm when parameterized by treewidth. In terms of kernelization, we show that, even when restricted to binary weights, Weighted Connected Matching does not admit a polynomial kernel when parameterized by vertex cover under standard complexity-theoretical hypotheses.

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Section: Introductionmentioning

confidence: 99%

Weighted Connected Matchings

Gomes¹,

Masquio²,

Pinto³

et al. 2022

Preprint

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show abstract

“…Matchings are a widely studied subject both in structural and algorithmic graph theory [1,2,3,4,5,6,7]. A matching is a subset M ⊆ E of edges of a graph G = (V, E) that do not share any endpoint.…”

Section: Introductionmentioning

confidence: 99%

Disconnected Matchings

Gomes¹,

Masquio²,

Pinto³

et al. 2021

Preprint

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“…Here, we will improve this result and prove that the problem remains NP-complete for planar perfect elimination bipartite graphs with maximum degree three and girth at least l, for every fixed integer l ≥ 3 (see Theorem 1). In [25], Panda and Chaudhary proved that the Acyclic Matching problem is NP-complete for comb-convex bipartite graphs and dually chordal graphs. They also proved that the problem is hard to approximate within a ratio of n 1− for any > 0, unless P = NP and this problem is APX-complete for 2k + 1-regular graphs for k ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

“…They also proved that the problem is hard to approximate within a ratio of n 1− for any > 0, unless P = NP and this problem is APX-complete for 2k + 1-regular graphs for k ≥ 3. The Acyclic Matching problem is known to be polynomial time solvable for the class of bipartite permutation graphs, chain graphs [26], P 4 −free graphs, 2P 3 −free graphs [13], chordal graphs [4], split graphs and proper interval graphs [25]. In [13], Furst and Rautenbach proved that for a given graph G, deciding that the size of the maximum acyclic matching of G is equal to the size of the maximum matching of G is NP-hard even for bipartite graphs with a perfect matching and the maximum degree 4.…”

Section: Introductionmentioning

confidence: 99%

On the Parameterized Complexity of the Acyclic Matching Problem

Hajebi¹,

Javadi²

2021

Preprint

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A matching is a set of edges in a graph with no common endpoint. A matching M is called acyclic if the induced subgraph on the endpoints of the edges in M is acyclic. Given a graph G and an integer k, Acyclic Matching Problem seeks for an acyclic matching of size k in G. The problem is known to be NP-complete. In this paper, we investigate the complexity of the problem in different aspects. First, we prove that the problem remains NP-complete for the class of planar bipartite graphs of maximum degree three and arbitrarily large girth. Also, the problem remains NP-complete for the class of planar line graphs with maximum degree four. Moreover, we study the parameterized complexity of the problem. In particular, we prove that the problem is W[1]-hard on bipartite graphs with respect to the parameter k. On the other hand, the problem is fixed parameter tractable with respect to k, for line graphs, C 4 -free graphs and every proper minor-closed class of graphs (including bounded tree-width and planar graphs).

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Acyclic Matching in Some Subclasses of Graphs

Cited by 13 publications

References 10 publications

Weighted Connected Matchings

Weighted Connected Matchings

Disconnected Matchings

On the Parameterized Complexity of the Acyclic Matching Problem

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