311 research outputs found

    Efficient and Perfect domination on circular-arc graphs

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    Given a graph G=(V,E)G = (V,E), a \emph{perfect dominating set} is a subset of vertices V′⊆V(G)V' \subseteq V(G) such that each vertex v∈V(G)∖V′v \in V(G)\setminus V' is dominated by exactly one vertex v′∈V′v' \in V'. An \emph{efficient dominating set} is a perfect dominating set V′V' where V′V' is also an independent set. These problems are usually posed in terms of edges instead of vertices. Both problems, either for the vertex or edge variant, remains NP-Hard, even when restricted to certain graphs families. We study both variants of the problems for the circular-arc graphs, and show efficient algorithms for all of them

    Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs

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    A path in an edge-colored graph GG is rainbow if no two edges of it are colored the same. The graph GG is rainbow-connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph GG is strongly rainbow-connected. The minimum number of colors needed to make GG rainbow-connected is known as the rainbow connection number of GG, and is denoted by rc(G)\text{rc}(G). Similarly, the minimum number of colors needed to make GG strongly rainbow-connected is known as the strong rainbow connection number of GG, and is denoted by src(G)\text{src}(G). We prove that for every k≥3k \geq 3, deciding whether src(G)≤k\text{src}(G) \leq k is NP-complete for split graphs, which form a subclass of chordal graphs. Furthermore, there exists no polynomial-time algorithm for approximating the strong rainbow connection number of an nn-vertex split graph with a factor of n1/2−ϵn^{1/2-\epsilon} for any ϵ>0\epsilon > 0 unless P = NP. We then turn our attention to block graphs, which also form a subclass of chordal graphs. We determine the strong rainbow connection number of block graphs, and show it can be computed in linear time. Finally, we provide a polynomial-time characterization of bridgeless block graphs with rainbow connection number at most 4.Comment: 13 pages, 3 figure

    Reduction Techniques for Graph Isomorphism in the Context of Width Parameters

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    We study the parameterized complexity of the graph isomorphism problem when parameterized by width parameters related to tree decompositions. We apply the following technique to obtain fixed-parameter tractability for such parameters. We first compute an isomorphism invariant set of potential bags for a decomposition and then apply a restricted version of the Weisfeiler-Lehman algorithm to solve isomorphism. With this we show fixed-parameter tractability for several parameters and provide a unified explanation for various isomorphism results concerned with parameters related to tree decompositions. As a possibly first step towards intractability results for parameterized graph isomorphism we develop an fpt Turing-reduction from strong tree width to the a priori unrelated parameter maximum degree.Comment: 23 pages, 4 figure

    A note on the minimum skew rank of a graph

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    The minimum skew rank mr−(F,G)mr^{-}(\mathbb{F},G) of a graph GG over a field F\mathbb{F} is the smallest possible rank among all skew symmetric matrices over F\mathbb{F}, whose (ii,jj)-entry (for i≠ji\neq j) is nonzero whenever ijij is an edge in GG and is zero otherwise. We give some new properties of the minimum skew rank of a graph, including a characterization of the graphs GG with cut vertices over the infinite field F\mathbb{F} such that mr−(F,G)=4mr^{-}(\mathbb{F},G)=4, determination of the minimum skew rank of kk-paths over a field F\mathbb{F}, and an extending of an existing result to show that mr−(F,G)=2match(G)=MR−(F,G)mr^{-}(\mathbb{F},G)=2match(G)=MR^{-}(\mathbb{F},G) for a connected graph GG with no even cycles and a field F\mathbb{F}, where match(G)match(G) is the matching number of GG, and MR−(F,G)MR^{-}(\mathbb{F},G) is the largest possible rank among all skew symmetric matrices over F\mathbb{F}
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