556 research outputs found

    Power domination on triangular grids

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    The concept of power domination emerged from the problem of monitoring electrical systems. Given a graph G and a set S \subseteq V (G), a set M of monitored vertices is built as follows: at first, M contains only the vertices of S and their direct neighbors, and then each time a vertex in M has exactly one neighbor not in M, this neighbor is added to M. The power domination number of a graph G is the minimum size of a set S such that this process ends up with the set M containing every vertex of G. We here show that the power domination number of a triangular grid T\_k with hexagonal-shape border of length k -- 1 is exactly $\lceil k/3 \rceil.Comment: Canadian Conference on Computational Geometry, Jul 2017, Ottawa, Canad

    The Price of Order

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    We present tight bounds on the spanning ratio of a large family of ordered θ\theta-graphs. A θ\theta-graph partitions the plane around each vertex into mm disjoint cones, each having aperture θ=2π/m\theta = 2 \pi/m. An ordered θ\theta-graph is constructed by inserting the vertices one by one and connecting each vertex to the closest previously-inserted vertex in each cone. We show that for any integer k1k \geq 1, ordered θ\theta-graphs with 4k+44k + 4 cones have a tight spanning ratio of 1+2sin(θ/2)/(cos(θ/2)sin(θ/2))1 + 2 \sin(\theta/2) / (\cos(\theta/2) - \sin(\theta/2)). We also show that for any integer k2k \geq 2, ordered θ\theta-graphs with 4k+24k + 2 cones have a tight spanning ratio of 1/(12sin(θ/2))1 / (1 - 2 \sin(\theta/2)). We provide lower bounds for ordered θ\theta-graphs with 4k+34k + 3 and 4k+54k + 5 cones. For ordered θ\theta-graphs with 4k+24k + 2 and 4k+54k + 5 cones these lower bounds are strictly greater than the worst case spanning ratios of their unordered counterparts. These are the first results showing that ordered θ\theta-graphs have worse spanning ratios than unordered θ\theta-graphs. Finally, we show that, unlike their unordered counterparts, the ordered θ\theta-graphs with 4, 5, and 6 cones are not spanners