1,122 research outputs found

    General dd-position sets

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    The general dd-position number gpd(G){\rm gp}_d(G) of a graph GG is the cardinality of a largest set SS for which no three distinct vertices from SS lie on a common geodesic of length at most dd. This new graph parameter generalizes the well studied general position number. We first give some results concerning the monotonic behavior of gpd(G){\rm gp}_d(G) with respect to the suitable values of dd. We show that the decision problem concerning finding gpd(G){\rm gp}_d(G) is NP-complete for any value of dd. The value of gpd(G){\rm gp}_d(G) when GG is a path or a cycle is computed and a structural characterization of general dd-position sets is shown. Moreover, we present some relationships with other topics including strong resolving graphs and dissociation sets. We finish our exposition by proving that gpd(G){\rm gp}_d(G) is infinite whenever GG is an infinite graph and dd is a finite integer.Comment: 16 page

    Kernelization and Parameterized Algorithms for 3-Path Vertex Cover

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    A 3-path vertex cover in a graph is a vertex subset CC such that every path of three vertices contains at least one vertex from CC. The parameterized 3-path vertex cover problem asks whether a graph has a 3-path vertex cover of size at most kk. In this paper, we give a kernel of 5k5k vertices and an O(1.7485k)O^*(1.7485^k)-time and polynomial-space algorithm for this problem, both new results improve previous known bounds.Comment: in TAMC 2016, LNCS 9796, 201

    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
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