Abstract:
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A [1, 2]-set S in a graph G is a vertex subset such that every vertex
not in S has at least one and at most two neighbors in it. If the additional
requirement that the set be independent is added, the existence of such
sets is not guaranteed in every graph. In this paper we provide local
conditions, depending on the degree of vertices, for the existence of
independent [1, 2]-sets in caterpillars. We also study the relationship
between independent [1, 2]-sets and independent dominating sets in this
graph class, that allows us to obtain an upper bound for the associated
parameter, the independent [1, 2]-number, in terms of the independent
domination number. |