The Parameterized Complexity of the Induced Matching Problem in Planar Graphs

Moser, Hannes; Sikdar, Somnath

doi:10.1007/978-3-540-73814-5_32

Cited by 17 publications

(20 citation statements)

References 40 publications

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“…These ideas were later abstracted by Guo and Niedermeier [15] who gave a framework to obtain linear kernels for planar graph problems possessing a certain "locality property". This framework has been successfully applied to yield linear kernels for the Connected Vertex Cover, Minimum Edge Dominating Set, Maximum Triangle Packing, Efficient Edge Dominating Set, Induced Matching and Full-Degree Spanning Tree problems [15,16,18]. However, the framework proposed by Guo and Niedermeier [15] in its current form is not able to handle problems like Feedback Vertex Set and Odd Cycle Transversal because these do not admit the "locality property" required by the framework.…”

Section: Introductionmentioning

confidence: 99%

Linear Kernel for Planar Connected Dominating Set

Lokshtanov

Mnich

Saurabh

2009

Lecture Notes in Computer Science

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We provide polynomial time data reduction rules for Connected Dominating Set on planar graphs and analyze these to obtain a linear kernel for the planar Connected Dominating Set problem. To obtain the desired kernel we introduce a method that we call reduce or refine. Our kernelization algorithm analyzes the input graph and either finds an appropriate reduction rule that can be applied, or zooms in on a region of the graph which is more amenable to reduction. We find this method of independent interest and believe that it will be useful to obtain linear kernels for other problems on planar graphs.

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Section: Introductionmentioning

confidence: 99%

Linear Kernel for Planar Connected Dominating Set

Lokshtanov

Mnich

Saurabh

2009

Lecture Notes in Computer Science

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“…This gives us the following theorem. We finish this section by noting that there is also an O * (4 t )-time algorithm by Moser and Sikdar [12], where t denotes the tree width of the input graph. It follows that our algorithm improves on this result, since for any graph G of at least 3 edges, bw(G) ≤ tw(G) + 1 ≤ 3…”

Section: Dynamic Programming On Graphs Of Bounded Branch Widthmentioning

confidence: 98%

“…For the class of planar graphs, Moser and Sikdar [12] showed that the problem has a linear kernel. The result of Kanj et al mentioned in Section 1.1 implies that the size of the kernel is bounded by 40k.…”

Section: Computational Perspectivementioning

confidence: 99%

Improved induced matchings in sparse graphs

Erman¹,

Kowalik²,

Krnc³

et al. 2010

Discrete Applied Mathematics

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a b s t r a c tAn induced matching in a graph G = (V , E) is a matching M such that (V , M) is an induced subgraph of G. Clearly, among two vertices with the same neighbourhood (called twins) at most one is matched in any induced matching, and if one of them is matched then there is another matching of the same size that matches the other vertex. Motivated by this, Kanj et al.[10] studied induced matchings in twinless graphs. They showed that any twinless planar graph contains an induced matching of size at least n 40 and that there are twinless planar graphs that do not contain an induced matching of size greater than n 27 + O(1). We improve both these bounds to n 28 + O(1), which is tight up to an additive constant. This implies that the problem of deciding whether a planar graph has an induced matching of size k has a kernel of size at most 28k. We also show for the first time that this problem is fixed parameter tractable for graphs of bounded arboricity. Kanj et al. also presented an algorithm which decides in O(2 159 √ k + n)-time whether an n-vertex planar graph contains an induced matching of size k. Our results improve the time complexity analysis of their algorithm. However, we also show a more efficient O(2 25.5 √ k + n)-time algorithm. Its main ingredient is a new, O * (4 l )-time algorithm for finding a maximum induced matching in a graph of branch width at most l.

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“…On the other hand, MIM has been shown to be solvable in polynomial time for several graph classes, including, for example, chordal graphs, circular arc graphs, weakly chordal graphs and outerplanar graphs (see [4,14] for a survey and [13] for hhd-free graphs). Recently, authors in [11] showed that planar twinless graphs always contain an induced matching of size at least n/40, while there are planar twinless graphs that do not contain an induced matching of size (n + 10)/27.…”

Section: Introductionmentioning

confidence: 99%

Maximum Induced Matching of Hexagonal Graphs

Erveš

Šparl

2015

Bull. Malays. Math. Sci. Soc.

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A matching in a graph is a set of edges no two of which share a common vertex. A matching is an induced matching if no two edges in the matching have a third edge in the graph connecting them. The problem of finding a maximum induced matching or shortly MIM is known to be NP-hard in general, and it remains so even when the input graph is bipartite. The decision problem of MIM is NP-complete in general, and it remains NP-complete even if restricted to several classes of graphs. On the other hand, the problem has been shown to be polynomial for some special sets of graphs. In this paper, we give tight upper and lower bounds on maximum induced matching in special subset of planar graphs, called hexagonal graphs.

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The Parameterized Complexity of the Induced Matching Problem in Planar Graphs

Cited by 17 publications

References 40 publications

Linear Kernel for Planar Connected Dominating Set

Linear Kernel for Planar Connected Dominating Set

Improved induced matchings in sparse graphs

Maximum Induced Matching of Hexagonal Graphs

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