130,841 research outputs found

    On the Complexity of Finding a Sun in a Graph

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    The sun is the graph obtained from a cycle of length even and at least six by adding edges to make the even-indexed vertices pairwise adjacent. Suns play an important role in the study of strongly chordal graphs. A graph is chordal if it does not contain an induced cycle of length at least four. A graph is strongly chordal if it is chordal and every even cycle has a chord joining vertices whose distance on the cycle is odd. Farber proved that a graph is strongly chordal if and only if it is chordal and contains no induced suns. There are well known polynomial-time algorithms for recognizing a sun in a chordal graph. Recently, polynomial-time algorithms for finding a sun for a larger class of graphs, the so-called HHD-free graphs (graphs containing no house, hole, or domino), have been discovered. In this paper, we prove the problem of deciding whether an arbitrary graph contains a sun is NP-complete

    Efficient motion planning for problems lacking optimal substructure

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    We consider the motion-planning problem of planning a collision-free path of a robot in the presence of risk zones. The robot is allowed to travel in these zones but is penalized in a super-linear fashion for consecutive accumulative time spent there. We suggest a natural cost function that balances path length and risk-exposure time. Specifically, we consider the discrete setting where we are given a graph, or a roadmap, and we wish to compute the minimal-cost path under this cost function. Interestingly, paths defined using our cost function do not have an optimal substructure. Namely, subpaths of an optimal path are not necessarily optimal. Thus, the Bellman condition is not satisfied and standard graph-search algorithms such as Dijkstra cannot be used. We present a path-finding algorithm, which can be seen as a natural generalization of Dijkstra's algorithm. Our algorithm runs in O((nBn)log(nBn)+nBm)O\left((n_B\cdot n) \log( n_B\cdot n) + n_B\cdot m\right) time, where~nn and mm are the number of vertices and edges of the graph, respectively, and nBn_B is the number of intersections between edges and the boundary of the risk zone. We present simulations on robotic platforms demonstrating both the natural paths produced by our cost function and the computational efficiency of our algorithm

    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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