672 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
A Greedy Partition Lemma for Directed Domination
A directed dominating set in a directed graph is a set of vertices of
such that every vertex has an adjacent vertex
in with directed to . The directed domination number of , denoted
by , is the minimum cardinality of a directed dominating set in .
The directed domination number of a graph , denoted , which is
the maximum directed domination number over all orientations of
. The directed domination number of a complete graph was first studied by
Erd\"{o}s [Math. Gaz. 47 (1963), 220--222], albeit in disguised form. In this
paper we prove a Greedy Partition Lemma for directed domination in oriented
graphs. Applying this lemma, we obtain bounds on the directed domination
number. In particular, if denotes the independence number of a graph
, we show that .Comment: 12 page
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