122,040 research outputs found
Zero forcing in iterated line digraphs
Zero forcing is a propagation process on a graph, or digraph, defined in
linear algebra to provide a bound for the minimum rank problem. Independently,
zero forcing was introduced in physics, computer science and network science,
areas where line digraphs are frequently used as models. Zero forcing is also
related to power domination, a propagation process that models the monitoring
of electrical power networks.
In this paper we study zero forcing in iterated line digraphs and provide a
relationship between zero forcing and power domination in line digraphs. In
particular, for regular iterated line digraphs we determine the minimum
rank/maximum nullity, zero forcing number and power domination number, and
provide constructions to attain them. We conclude that regular iterated line
digraphs present optimal minimum rank/maximum nullity, zero forcing number and
power domination number, and apply our results to determine those parameters on
some families of digraphs often used in applications
On the multipacking number of grid graphs
In 2001, Erwin introduced broadcast domination in graphs. It is a variant of
classical domination where selected vertices may have different domination
powers. The minimum cost of a dominating broadcast in a graph is denoted
. The dual of this problem is called multipacking: a multipacking
is a set of vertices such that for any vertex and any positive integer
, the ball of radius around contains at most vertices of .
The maximum size of a multipacking in a graph is denoted mp(G). Naturally
mp(G) . Earlier results by Farber and by Lubiw show that
broadcast and multipacking numbers are equal for strongly chordal graphs. In
this paper, we show that all large grids (height at least 4 and width at least
7), which are far from being chordal, have their broadcast and multipacking
numbers equal
On the Approximability of External-Influence-Driven Problems
Domination problems in general can capture situations in which some entities
have an effect on other entities (and sometimes on themselves). The usual goal
is to select a minimum number of entities that can influence a target group of
entities or to influence a maximum number of target entities with a certain
number of available influencers. In this work, we focus on the distinction
between \textit{internal} and \textit{external} domination in the respective
maximization problem. In particular, a dominator can dominate its entire
neighborhood in a graph, internally dominating itself, while those of its
neighbors which are not dominators themselves are externally dominated. We
study the problem of maximizing the external domination that a given number of
dominators can yield and we present a 0.5307-approximation algorithm for this
problem. Moreover, our methods provide a framework for approximating a number
of problems that can be cast in terms of external domination. In particular, we
observe that an interesting interpretation of the maximum coverage problem can
capture a new problem in elections, in which we want to maximize the number of
\textit{externally represented} voters. We study this problem in two different
settings, namely Non-Secrecy and Rational-Candidate, and provide
approximability analysis for two alternative approaches; our analysis reveals,
among other contributions, that an earlier resource allocation algorithm is, in
fact, a 0.462-approximation algorithm for maximum external domination in
directed graphs
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