5,310 research outputs found
Distributed Dominating Sets on Grids
This paper presents a distributed algorithm for finding near optimal
dominating sets on grids. The basis for this algorithm is an existing
centralized algorithm that constructs dominating sets on grids. The size of the
dominating set provided by this centralized algorithm is upper-bounded by
for grids and its difference
from the optimal domination number of the grid is upper-bounded by five. Both
the centralized and distributed algorithms are generalized for the -distance
dominating set problem, where all grid vertices are within distance of the
vertices in the dominating set.Comment: 10 pages, 9 figures, accepted in ACC 201
On (t,r) Broadcast Domination Numbers of Grids
The domination number of a graph is the minimum cardinality of
any subset such that every vertex in is in or adjacent to
an element of . Finding the domination numbers of by grids was an
open problem for nearly 30 years and was finally solved in 2011 by Goncalves,
Pinlou, Rao, and Thomass\'e. Many variants of domination number on graphs have
been defined and studied, but exact values have not yet been obtained for
grids. We will define a family of domination theories parameterized by pairs of
positive integers where which generalize domination
and distance domination theories for graphs. We call these domination numbers
the broadcast domination numbers. We give the exact values of
broadcast domination numbers for small grids, and we identify upper bounds for
the broadcast domination numbers for large grids and conjecture that
these bounds are tight for sufficiently large grids.Comment: 28 pages, 43 figure
Protecting a Graph with Mobile Guards
Mobile guards on the vertices of a graph are used to defend it against
attacks on either its vertices or its edges. Various models for this problem
have been proposed. In this survey we describe a number of these models with
particular attention to the case when the attack sequence is infinitely long
and the guards must induce some particular configuration before each attack,
such as a dominating set or a vertex cover. Results from the literature
concerning the number of guards needed to successfully defend a graph in each
of these problems are surveyed.Comment: 29 pages, two figures, surve
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