13 research outputs found

    29th International Symposium on Algorithms and Computation: ISAAC 2018, December 16-19, 2018, Jiaoxi, Yilan, Taiwan

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    Edge-Stable Equimatchable Graphs

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    A graph GG is \emph{equimatchable} if every maximal matching of GG has the same cardinality. We are interested in equimatchable graphs such that the removal of any edge from the graph preserves the equimatchability. We call an equimatchable graph GG \emph{edge-stable} if GeG\setminus {e}, that is the graph obtained by the removal of edge ee from GG, is also equimatchable for any eE(G)e \in E(G). After noticing that edge-stable equimatchable graphs are either 2-connected factor-critical or bipartite, we characterize edge-stable equimatchable graphs. This characterization yields an O(min(n3.376,n1.5m))O(\min(n^{3.376}, n^{1.5}m)) time recognition algorithm. Lastly, we introduce and shortly discuss the related notions of edge-critical, vertex-stable and vertex-critical equimatchable graphs. In particular, we emphasize the links between our work and the well-studied notion of shedding vertices, and point out some open questions

    On Minimum Maximal Distance-k Matchings

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    We study the computational complexity of several problems connected with finding a maximal distance-kk matching of minimum cardinality or minimum weight in a given graph. We introduce the class of kk-equimatchable graphs which is an edge analogue of kk-equipackable graphs. We prove that the recognition of kk-equimatchable graphs is co-NP-complete for any fixed k2k \ge 2. We provide a simple characterization for the class of strongly chordal graphs with equal kk-packing and kk-domination numbers. We also prove that for any fixed integer 1\ell \ge 1 the problem of finding a minimum weight maximal distance-22\ell matching and the problem of finding a minimum weight (21)(2 \ell - 1)-independent dominating set cannot be approximated in polynomial time in chordal graphs within a factor of δlnV(G)\delta \ln |V(G)| unless P=NP\mathrm{P} = \mathrm{NP}, where δ\delta is a fixed constant (thereby improving the NP-hardness result of Chang for the independent domination case). Finally, we show the NP-hardness of the minimum maximal induced matching and independent dominating set problems in large-girth planar graphs.Comment: 15 pages, 4 figure

    Weighted Well-Covered Claw-Free Graphs

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    A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space. Given an input claw-free graph G, we present an O(n^6)algortihm, whose input is a claw-free graph G, and output is the vector space of weight functions w, for which G is w-well-covered. A graph G is equimatchable if all its maximal matchings are of the same cardinality. Assume that a weight function w is defined on the edges of G. Then G is w-equimatchable if all its maximal matchings are of the same weight. For every graph G, the set of weight functions w such that G is w-equimatchable is a vector space. We present an O(m*n^4 + n^5*log(n)) algorithm which receives an input graph G, and outputs the vector space of weight functions w such that G is w-equimatchable.Comment: 14 pages, 1 figur

    The number of maximum matchings in a tree

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    We determine upper and lower bounds for the number of maximum matchings (i.e., matchings of maximum cardinality) m(T)m(T) of a tree TT of given order. While the trees that attain the lower bound are easily characterised, the trees with largest number of maximum matchings show a very subtle structure. We give a complete characterisation of these trees and derive that the number of maximum matchings in a tree of order nn is at most O(1.391664n)O(1.391664^n) (the precise constant being an algebraic number of degree 14). As a corollary, we improve on a recent result by G\'orska and Skupie\'n on the number of maximal matchings (maximal with respect to set inclusion).Comment: 38 page

    On the algorithmic complexity of twelve covering and independence parameters of graphs

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    The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs
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