Improved induced matchings in sparse graphs

Abstract

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.

Essentially Tight Kernels for (Weakly) Closed Graphs

“…We then show that this strategy helps in obtaining kernels on weakly closed graphs for Capacitated Vertex Cover, Connected Vertex Cover, and Induced Matching all parameterized by the natural parameter solution size k. For these problems, polynomial kernels in degenerate graphs were known [5,6,12,18]. Our results thus extend the class of graphs for which polynomial kernels are known for these problems.…”

“…For c-closed graphs, we recently developed a kernel with O(c 7 k 8 ) vertices [22]. As for d-degenerate graphs, Kanj et al [18] and Erman et al [12] independently presented kernels of size k O(d) . Later, Cygan et al [5] provided a matching lower bound k o(d) on the kernel size.…”

“…In a nutshell, we show that the application of a series of reduction rules results in a graph with a (γk) O(1) -size vertex cover. We do so by combining different known kernelizations, namely, the one of Erman et al [12] with the one in our previous work [22]. Lemma 2.2 and the reduction rule which removes twin vertices then gives us a kernel of size (γk) O(γ) .…”

Tight Kernel Bounds for Problems on Graphs with Small Degeneracy

“…Arguably the most important kernelization result in d-degenerate graphs is due to Philip et al [29] who showed a O(k (d+1) 2 ) size kernel for Dominating Set, and an O(k d+1 ) size kernel for Independent Dominating Set. Erman et al [14] and Kanj et al [24] independently gave a O(k d ) kernel for the Induced Matching problem, while Cygan et al [8] showed a O(k d+1 ) kernel is for Connected Vertex Cover. While all these results give polynomial kernels, the exponent of the polynomial depends on d, leaving open the question of kernels of polynomial size with a fixed constant degree.…”

Towards optimal kernel for connected vertex cover in planar graphs

“…Motivated by this, researchers seek for linear kernels with constants as small as possible. For example, by now there is known a 67k-vertex kernel for Dominating Set [3], a 28k-vertex kernel for Induced Matching [8] or a 75k-vertex kernel for Maximum Triangle Packing [17].…”