1,551 research outputs found
Power domination in maximal planar graphs
Power domination in graphs emerged from the problem of monitoring an
electrical system by placing as few measurement devices in the system as
possible. It corresponds to a variant of domination that includes the
possibility of propagation. For measurement devices placed on a set S of
vertices of a graph G, the set of monitored vertices is initially the set S
together with all its neighbors. Then iteratively, whenever some monitored
vertex v has a single neighbor u not yet monitored, u gets monitored. A set S
is said to be a power dominating set of the graph G if all vertices of G
eventually are monitored. The power domination number of a graph is the minimum
size of a power dominating set. In this paper, we prove that any maximal planar
graph of order n 6 admits a power dominating set of size at most (n--2)/4
Reconfiguration on sparse graphs
A vertex-subset graph problem Q defines which subsets of the vertices of an
input graph are feasible solutions. A reconfiguration variant of a
vertex-subset problem asks, given two feasible solutions S and T of size k,
whether it is possible to transform S into T by a sequence of vertex additions
and deletions such that each intermediate set is also a feasible solution of
size bounded by k. We study reconfiguration variants of two classical
vertex-subset problems, namely Independent Set and Dominating Set. We denote
the former by ISR and the latter by DSR. Both ISR and DSR are PSPACE-complete
on graphs of bounded bandwidth and W[1]-hard parameterized by k on general
graphs. We show that ISR is fixed-parameter tractable parameterized by k when
the input graph is of bounded degeneracy or nowhere-dense. As a corollary, we
answer positively an open question concerning the parameterized complexity of
the problem on graphs of bounded treewidth. Moreover, our techniques generalize
recent results showing that ISR is fixed-parameter tractable on planar graphs
and graphs of bounded degree. For DSR, we show the problem fixed-parameter
tractable parameterized by k when the input graph does not contain large
bicliques, a class of graphs which includes graphs of bounded degeneracy and
nowhere-dense graphs
Bounding the Porous Exponential Domination Number of Apollonian Networks
Given a graph G with vertex set V, a subset S of V is a dominating set if
every vertex in V is either in S or adjacent to some vertex in S. The size of a
smallest dominating set is called the domination number of G. We study a
variant of domination called porous exponential domination in which each vertex
v of V is assigned a weight by each vertex s of S that decreases exponentially
as the distance between v and s increases. S is a porous exponential dominating
set for G if all vertices in S distribute to vertices in G a total weight of at
least 1. The porous exponential domination number of G is the size of a
smallest porous exponential dominating set. In this paper we compute bounds for
the porous exponential domination number of special graphs known as Apollonian
networks.Comment: 8 pages, 5 figures, 1 table, Research partially funded by CURM, the
Center for Undergraduate Research, and NSF grant DMS-114869
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
- …