Transversal total hop domination in graphs

Author:

Bonsocan Maria Andrea O.1ORCID,Jamil Ferdinand P.2ORCID

Affiliation:

1. Department of Mathematics and Statistics, CSM, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines

2. Department of Mathematics and Statistics, CSM, Center for Mathematical and Theoretical Physical Sciences, PRISM, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines

Abstract

Let [Formula: see text] be a finite simple graph. A set [Formula: see text] is a total hop dominating set of [Formula: see text] if for every [Formula: see text], there exists [Formula: see text] such that [Formula: see text]. Any total hop dominating set of [Formula: see text] of minimum cardinality is a [Formula: see text]-set of [Formula: see text]. A total hop dominating set [Formula: see text] of [Formula: see text] which intersects every [Formula: see text]-set of [Formula: see text] is a transversal total hop dominating set. The minimum cardinality [Formula: see text] of a transversal total hop dominating set in [Formula: see text] is the transversal total hop domination number of [Formula: see text]. In this paper, we initiate the study of transversal total hop domination in graphs. First, we characterize a graph [Formula: see text] of order [Formula: see text] for which [Formula: see text] is [Formula: see text] or [Formula: see text], and also we determine the specific values of [Formula: see text] for some special graphs [Formula: see text]. Next, we solve some realization problems involving [Formula: see text] with other parameters of [Formula: see text]. Finally, we investigate the transversal total hop domination in the complementary prism, corona, and lexicographic product of graphs.

Funder

Department of Science and Technology, Philippines

Publisher

World Scientific Pub Co Pte Ltd

Subject

Discrete Mathematics and Combinatorics

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