136 research outputs found
Defensive alliances in graphs: a survey
A set of vertices of a graph is a defensive -alliance in if
every vertex of has at least more neighbors inside of than outside.
This is primarily an expository article surveying the principal known results
on defensive alliances in graph. Its seven sections are: Introduction,
Computational complexity and realizability, Defensive -alliance number,
Boundary defensive -alliances, Defensive alliances in Cartesian product
graphs, Partitioning a graph into defensive -alliances, and Defensive
-alliance free sets.Comment: 25 page
The k-metric dimension of a graph
As a generalization of the concept of a metric basis, this article introduces
the notion of -metric basis in graphs. Given a connected graph , a
set is said to be a -metric generator for if the elements
of any pair of different vertices of are distinguished by at least
elements of , i.e., for any two different vertices , there exist
at least vertices such that for every . A metric generator of minimum
cardinality is called a -metric basis and its cardinality the -metric
dimension of . A connected graph is -metric dimensional if is the
largest integer such that there exists a -metric basis for . We give a
necessary and sufficient condition for a graph to be -metric dimensional and
we obtain several results on the -metric dimension
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