67 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
Routing on the Visibility Graph
We consider the problem of routing on a network in the presence of line
segment constraints (i.e., obstacles that edges in our network are not allowed
to cross). Let be a set of points in the plane and let be a set of
non-crossing line segments whose endpoints are in . We present two
deterministic 1-local -memory routing algorithms that are guaranteed to
find a path of at most linear size between any pair of vertices of the
\emph{visibility graph} of with respect to a set of constraints (i.e.,
the algorithms never look beyond the direct neighbours of the current location
and store only a constant amount of additional information). Contrary to {\em
all} existing deterministic local routing algorithms, our routing algorithms do
not route on a plane subgraph of the visibility graph. Additionally, we provide
lower bounds on the routing ratio of any deterministic local routing algorithm
on the visibility graph.Comment: An extended abstract of this paper appeared in the proceedings of the
28th International Symposium on Algorithms and Computation (ISAAC 2017).
Final version appeared in the Journal of Computational Geometr
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