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Total domination excellent trees
AbstractA set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S (other than itself). The graph G is called total domination excellent if every vertex belongs to some total dominating set of G of minimum cardinality. We provide a constructive characterization of total domination excellent trees
Protecting a Graph with Mobile Guards
Mobile guards on the vertices of a graph are used to defend it against
attacks on either its vertices or its edges. Various models for this problem
have been proposed. In this survey we describe a number of these models with
particular attention to the case when the attack sequence is infinitely long
and the guards must induce some particular configuration before each attack,
such as a dominating set or a vertex cover. Results from the literature
concerning the number of guards needed to successfully defend a graph in each
of these problems are surveyed.Comment: 29 pages, two figures, surve
A characterization of trees with equal 2-domination and 2-independence numbers
A set of vertices in a graph is a -dominating set if every vertex
of not in is adjacent to at least two vertices in , and is a
-independent set if every vertex in is adjacent to at most one vertex of
. The -domination number is the minimum cardinality of a
-dominating set in , and the -independence number is the
maximum cardinality of a -independent set in . Chellali and Meddah [{\it
Trees with equal -domination and -independence numbers,} Discussiones
Mathematicae Graph Theory 32 (2012), 263--270] provided a constructive
characterization of trees with equal -domination and -independence
numbers. Their characterization is in terms of global properties of a tree, and
involves properties of minimum -dominating and maximum -independent sets
in the tree at each stage of the construction. We provide a constructive
characterization that relies only on local properties of the tree at each stage
of the construction.Comment: 17 pages, 4 figure
Trees with Maximum p-Reinforcement Number
Let be a graph and a positive integer. The -domination
number \g_p(G) is the minimum cardinality of a set with
for all . The -reinforcement
number is the smallest number of edges whose addition to results
in a graph with \g_p(G')<\g_p(G). Recently, it was proved by Lu et al.
that for a tree and . In this paper, we
characterize all trees attaining this upper bound for
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