10 research outputs found
Algorithms for the Global Domination Problem
A dominating set D in a graph G is a subset of its vertices such that every
vertex of the graph which does not belong to set D is adjacent to at least one
vertex from set D. A set of vertices of graph G is a global dominating set if
it is a dominating set for both, graph G and its complement. The objective is
to find a global dominating set with the minimum cardinality. The problem is
known to be NP-hard. Neither exact nor approximation algorithm existed . We
propose two exact solution methods, one of them being based on an integer
linear program (ILP) formulation, three heuristic algorithms and a special
purification procedure that further reduces the size of a global dominated set
delivered by any of our heuristic algorithms. We show that the problem remains
NP-hard for restricted types of graphs and specify some families of graphs for
which the heuristics guarantee the optimality. The second exact algorithm
turned out to be about twice faster than ILP for graphs with more than 230
vertices and up to 1080 vertices, which were the largest benchmark instances
that were solved optimally. The heuristics were tested for the existing 2284
benchmark problem instances with up to 14000 vertices and delivered solutions
for the largest instances in less than one minute. Remarkably, for about 52% of
the 1000 instances with the obtained optimal solutions, at least one of the
heuristics generated an optimal solution, where the average approximation error
for the remaining instances was 1.07%
Revan-degree indices on random graphs
Given a simple connected non-directed graph , we consider two
families of graph invariants:
(which has gained interest recently) and (that we introduce in this work); where denotes the edge of
connecting the vertices and , is the Revan degree of the
vertex , and is a function of the Revan vertex degrees. Here, with and the maximum and minimum
degrees among the vertices of and is the degree of the vertex .
Particularly, we apply both and R on two models of
random graphs: Erd\"os-R\'enyi graphs and random geometric graphs. By a
thorough computational study we show that \left and
\left, normalized to the order of the graph, scale
with the average Revan degree \left; here \left
denotes the average over an ensemble of random graphs. Moreover, we provide
analytical expressions for several graph invariants of both families in the
dense graph limit.Comment: 16 pages, 10 figure
Spectral and localization properties of random bipartite graphs
Bipartite graphs are often found to represent the connectivity between the
components of many systems such as ecosystems. A bipartite graph is a set of
nodes that is decomposed into two disjoint subsets, having and
vertices each, such that there are no adjacent vertices within the same set.
The connectivity between both sets, which is the relevant quantity in terms of
connections, can be quantified by a parameter that equals the
ratio of existent adjacent pairs over the total number of possible adjacent
pairs. Here, we study the spectral and localization properties of such random
bipartite graphs. Specifically, within a Random Matrix Theory (RMT) approach,
we identify a scaling parameter that fixes the
localization properties of the eigenvectors of the adjacency matrices of random
bipartite graphs. We also show that, when ) the
eigenvectors are localized (extended), whereas the
localization--to--delocalization transition occurs in the interval
. Finally, given the potential applications of our findings, we
round off the study by demonstrating that for fixed , the spectral
properties of our graph model are also universal.Comment: 17 pages, 10 figure