556 research outputs found
Power domination on triangular grids
The concept of power domination emerged from the problem of monitoring
electrical systems. Given a graph G and a set S 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
We present tight bounds on the spanning ratio of a large family of ordered
-graphs. A -graph partitions the plane around each vertex into
disjoint cones, each having aperture . An ordered
-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 , ordered -graphs with
cones have a tight spanning ratio of . We also show that for any integer , ordered
-graphs with cones have a tight spanning ratio of . We provide lower bounds for ordered -graphs with and cones. For ordered -graphs with and
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 -graphs have worse spanning ratios than unordered
-graphs. Finally, we show that, unlike their unordered counterparts,
the ordered -graphs with 4, 5, and 6 cones are not spanners
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