Disjunctive total domination in graphs

Michael A. Henning () and Viroshan Naicker ()
Additional contact information
Michael A. Henning: University of Johannesburg
Viroshan Naicker: University of Johannesburg

Journal of Combinatorial Optimization, 2016, vol. 31, issue 3, No 10, 1090-1110

Abstract: Abstract Let $$G$$ G be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, $$\gamma _t(G)$$ γ t ( G ) . A set $$S$$ S of vertices in $$G$$ G is a disjunctive total dominating set of $$G$$ G if every vertex is adjacent to a vertex of $$S$$ S or has at least two vertices in $$S$$ S at distance $$2$$ 2 from it. The disjunctive total domination number, $$\gamma ^d_t(G)$$ γ t d ( G ) , is the minimum cardinality of such a set. We observe that $$\gamma ^d_t(G) \le \gamma _t(G)$$ γ t d ( G ) ≤ γ t ( G ) . We prove that if $$G$$ G is a connected graph of order $$n \ge 8$$ n ≥ 8 , then $$\gamma ^d_t(G) \le 2(n-1)/3$$ γ t d ( G ) ≤ 2 ( n - 1 ) / 3 and we characterize the extremal graphs. It is known that if $$G$$ G is a connected claw-free graph of order $$n$$ n , then $$\gamma _t(G) \le 2n/3$$ γ t ( G ) ≤ 2 n / 3 and this upper bound is tight for arbitrarily large $$n$$ n . We show this upper bound can be improved significantly for the disjunctive total domination number. We show that if $$G$$ G is a connected claw-free graph of order $$n > 14$$ n > 14 , then $$\gamma ^d_t(G) \le 4n/7$$ γ t d ( G ) ≤ 4 n / 7 and we characterize the graphs achieving equality in this bound.

Keywords: Total dominating set; Disjunctive total dominating set; Claw-free; 05C69 (search for similar items in EconPapers)
Date: 2016
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://link.springer.com/10.1007/s10878-014-9811-4 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jcomop:v:31:y:2016:i:3:d:10.1007_s10878-014-9811-4

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10878

DOI: 10.1007/s10878-014-9811-4

Access Statistics for this article

Journal of Combinatorial Optimization is currently edited by Thai, My T.

More articles in Journal of Combinatorial Optimization from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().