Isolation Number versus Domination Number of Trees

Magdalena Lemańska, María José Souto-Salorio, Adriana Dapena and Francisco J. Vazquez-Araujo
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Magdalena Lemańska: Department of Technical Physics and Applied Mathematics, Gdansk University of Technology, ul. Narutowicza 11/12, 80-233 Gdansk, Poland
María José Souto-Salorio: Differential Geometry and Its Applcations Research Group, University of A Coruña, Campus de Elviña, 15071 A Coruña, Spain
Adriana Dapena: CITIC Research Center, University of A Coruña, Campus de Elviña, 15071 A Coruña, Spain
Francisco J. Vazquez-Araujo: CITIC Research Center, University of A Coruña, Campus de Elviña, 15071 A Coruña, Spain

Mathematics, 2021, vol. 9, issue 12, 1-10

Abstract: If G = ( V G , E G ) is a graph of order n , we call S ? V G an isolating set if the graph induced by V G ? N G [ S ] contains no edges. The minimum cardinality of an isolating set of G is called the isolation number of G , and it is denoted by ? ( G ) . It is known that ? ( G ) ? n 3 and the bound is sharp. A subset S ? V G is called dominating in G if N G [ S ] = V G . The minimum cardinality of a dominating set of G is the domination number, and it is denoted by ? ( G ) . In this paper, we analyze a family of trees T where ? ( T ) = ? ( T ) , and we prove that ? ( T ) = n 3 implies ? ( T ) = ? ( T ) . Moreover, we give different equivalent characterizations of such graphs and we propose simple algorithms to build these trees from the connections of stars.

Keywords: domination number; isolation number; trees; algorithms (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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