Excellent graphs with respect to domination: subgraphs induced by minimum dominating sets release_qdopeiq6gveybf2b3nkjed2pfy

by Vladimir Samodivkin

Released as a article .

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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Date   2020-10-07
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arXiv  2010.03219v1
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