5,825 research outputs found
Exponential Domination in Subcubic Graphs
As a natural variant of domination in graphs, Dankelmann et al. [Domination
with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduce
exponential domination, where vertices are considered to have some dominating
power that decreases exponentially with the distance, and the dominated
vertices have to accumulate a sufficient amount of this power emanating from
the dominating vertices. More precisely, if is a set of vertices of a graph
, then is an exponential dominating set of if for every vertex
in , where is the distance
between and in the graph . The exponential domination number of is the minimum
order of an exponential dominating set of .
In the present paper we study exponential domination in subcubic graphs. Our
results are as follows: If is a connected subcubic graph of order ,
then For every , there is some such that
for every cubic graph of girth at least
. For every , there are infinitely many cubic
graphs with . If is a
subcubic tree, then For a given subcubic
tree, can be determined in polynomial time. The minimum
exponential dominating set problem is APX-hard for subcubic graphs
Limited packings of closed neighbourhoods in graphs
The k-limited packing number, , of a graph , introduced by
Gallant, Gunther, Hartnell, and Rall, is the maximum cardinality of a set
of vertices of such that every vertex of has at most elements of
in its closed neighbourhood. The main aim in this paper is to prove the
best-possible result that if is a cubic graph, then , improving the previous lower bound given by Gallant, \emph{et al.}
In addition, we construct an infinite family of graphs to show that lower
bounds given by Gagarin and Zverovich are asymptotically best-possible, up to a
constant factor, when is fixed and tends to infinity. For
tending to infinity and tending to infinity sufficiently
quickly, we give an asymptotically best-possible lower bound for ,
improving previous bounds
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
Greed is Good for Deterministic Scale-Free Networks
Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. In fact, the behavior of real-world networks and random graph models can be the complete opposite of one another, depending on the considered property. Brach, Cygan, Lacki, and Sankowski [SODA 2016] introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U) and (2) power-law neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLB-N). They showed that many real-world networks satisfy both deterministic properties and exploit them to design faster algorithms for a number of classical graph problems like transitive closure, maximum matching, determinant, PageRank, matrix inverse, counting triangles and maximum clique.
We complement the work of Brach et al. by showing that some well-studied random graph models exhibit both the mentioned PLB properties and additionally also a power-law lower bound on the degree distribution (PLB-L). All three properties hold with high probability for Chung-Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. As a consequence, all results of Brach et al. also hold with high probability for Chung-Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs.
In the second part of this work we study three classical NP-hard combinatorial optimization problems on PLB networks. It is known that on general graphs, a greedy algorithm, which chooses nodes in the order of their degree, only achieves an approximation factor of asymptotically at least logarithmic in the maximum degree for Minimum Vertex Cover and Minimum Dominating Set, and an approximation factor of asymptotically at least the maximum degree for Maximum Independent Set. We prove that the PLB-U property suffices such that the greedy approach achieves a constant-factor approximation for all three problems. We also show that all three combinatorial optimization problems are APX-complete, even if all PLB-properties hold. Hence, a PTAS cannot be expected, unless P=NP
Generation of cubic graphs and snarks with large girth
We describe two new algorithms for the generation of all non-isomorphic cubic
graphs with girth at least which are very efficient for
and show how these algorithms can be efficiently restricted to generate snarks
with girth at least .
Our implementation of these algorithms is more than 30, respectively 40 times
faster than the previously fastest generator for cubic graphs with girth at
least 6 and 7, respectively.
Using these generators we have also generated all non-isomorphic snarks with
girth at least 6 up to 38 vertices and show that there are no snarks with girth
at least 7 up to 42 vertices. We present and analyse the new list of snarks
with girth 6.Comment: 27 pages (including appendix
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