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Disjunctive Total Domination in Graphs
Let be a graph with no isolated vertex. In this paper, we study a
parameter that is a relaxation of arguably the most important domination
parameter, namely the total domination number, . A set of
vertices in is a disjunctive total dominating set of if every vertex is
adjacent to a vertex of or has at least two vertices in at distance2
from it. The disjunctive total domination number, , is the
minimum cardinality of such a set. We observe that . We prove that if is a connected graph of order, then
and we characterize the extremal graphs. It is
known that if is a connected claw-free graph of order, then and this upper bound is tight for arbitrarily large. We show this
upper bound can be improved significantly for the disjunctive total domination
number. We show that if is a connected claw-free graph of order,
then and we characterize the graphs achieving equality
in this bound.Comment: 23 page
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