32,901 research outputs found
Counting dominating sets and related structures in graphs
We consider some problems concerning the maximum number of (strong)
dominating sets in a regular graph, and their weighted analogues. Our primary
tool is Shearer's entropy lemma. These techniques extend to a reasonably broad
class of graph parameters enumerating vertex colorings satisfying conditions on
the multiset of colors appearing in (closed) neighborhoods. We also generalize
further to enumeration problems for what we call existence homomorphisms. Here
our results are substantially less complete, though we do solve some natural
problems
Reconfiguration of Dominating Sets
We explore a reconfiguration version of the dominating set problem, where a
dominating set in a graph is a set of vertices such that each vertex is
either in or has a neighbour in . In a reconfiguration problem, the goal
is to determine whether there exists a sequence of feasible solutions
connecting given feasible solutions and such that each pair of
consecutive solutions is adjacent according to a specified adjacency relation.
Two dominating sets are adjacent if one can be formed from the other by the
addition or deletion of a single vertex.
For various values of , we consider properties of , the graph
consisting of a vertex for each dominating set of size at most and edges
specified by the adjacency relation. Addressing an open question posed by Haas
and Seyffarth, we demonstrate that is not necessarily
connected, for the maximum cardinality of a minimal dominating set
in . The result holds even when graphs are constrained to be planar, of
bounded tree-width, or -partite for . Moreover, we construct an
infinite family of graphs such that has exponential
diameter, for the minimum size of a dominating set. On the positive
side, we show that is connected and of linear diameter for any
graph on vertices having at least independent edges.Comment: 12 pages, 4 figure
Dominating sets in projective planes
We describe small dominating sets of the incidence graphs of finite
projective planes by establishing a stability result which shows that
dominating sets are strongly related to blocking and covering sets. Our main
result states that if a dominating set in a projective plane of order is
smaller than (i.e., twice the size of a Baer subplane), then
it contains either all but possibly one points of a line or all but possibly
one lines through a point. Furthermore, we completely characterize dominating
sets of size at most . In Desarguesian planes, we could rely on
strong stability results on blocking sets to show that if a dominating set is
sufficiently smaller than 3q, then it consists of the union of a blocking set
and a covering set apart from a few points and lines.Comment: 19 page
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
Distributed Dominating Set Approximations beyond Planar Graphs
The Minimum Dominating Set (MDS) problem is one of the most fundamental and
challenging problems in distributed computing. While it is well-known that
minimum dominating sets cannot be approximated locally on general graphs, over
the last years, there has been much progress on computing local approximations
on sparse graphs, and in particular planar graphs.
In this paper we study distributed and deterministic MDS approximation
algorithms for graph classes beyond planar graphs. In particular, we show that
existing approximation bounds for planar graphs can be lifted to bounded genus
graphs, and present (1) a local constant-time, constant-factor MDS
approximation algorithm and (2) a local -time
approximation scheme. Our main technical contribution is a new analysis of a
slightly modified variant of an existing algorithm by Lenzen et al.
Interestingly, unlike existing proofs for planar graphs, our analysis does not
rely on direct topological arguments.Comment: arXiv admin note: substantial text overlap with arXiv:1602.0299
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