Excellent graphs with respect to domination: subgraphs induced by minimum dominating sets
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by
Vladimir Samodivkin
2020
Abstract
A graph G=(V,E) is γ-excellent if V is a union of all
γ-sets of G, where γ stands for the domination number. Let
ℐ be a set of all mutually nonisomorphic graphs and ∅≠ℋ⊊ℐ. In this paper we initiate the study
of the ℋ-γ-excellent graphs, which we define as follows. A
graph G is ℋ-γ-excellent if the following hold: (i) for
every H ∈ℋ and for each x ∈ V(G) there exists an induced
subgraph H_x of G such that H and H_x are isomorphic, x ∈ V(H_x)
and V(H_x) is a subset of some γ-set of G, and (ii) the vertex set
of every induced subgraph H of G, which is isomorphic to some element of
ℋ, is a subset of some γ-set of G. For each of some well
known graphs, including cycles, trees and some cartesian products of two
graphs, we describe its largest set ℋ⊊ℐ for
which the graph is ℋ-γ-excellent. Results on
γ-excellent regular graphs and a generalized lexicographic product of
graphs are presented. Several open problems and questions are posed.
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2010.03219v1
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