A new lower bound on the domination number of a graph

Majid Hajian (), Michael A. Henning () and Nader Jafari Rad ()
Additional contact information
Majid Hajian: Shahrood University of Technology
Michael A. Henning: University of Johannesburg
Nader Jafari Rad: Shahed University

Journal of Combinatorial Optimization, 2019, vol. 38, issue 3, No 6, 738 pages

Abstract: Abstract A set S of vertices in a graph G is a dominating set of G if every vertex not in S is adjacent to some vertex in S. The domination number, $$\gamma (G)$$ γ ( G ) , of G is the minimum cardinality of a dominating set of G. Lemańska (Discuss Math Graph Theory 24:165–170, 2004) showed that if T is a tree of order $$n \ge 2$$ n ≥ 2 with $$\ell $$ ℓ leaves, then $$\gamma (T) \ge (n-\ell +2)/3$$ γ ( T ) ≥ ( n - ℓ + 2 ) / 3 , and characterized all trees achieving equality in this bound. In this paper, we first characterize all trees T of order n with $$\ell $$ ℓ leaves satisfying $$\gamma (T) = \lceil (n - \ell + 2)/3 \rceil $$ γ ( T ) = ⌈ ( n - ℓ + 2 ) / 3 ⌉ . We then generalize this result to connected graphs and show that if G is a connected graph of order $$n \ge 2$$ n ≥ 2 with $$k \ge 0$$ k ≥ 0 cycles and $$\ell $$ ℓ leaves, then $$\gamma (G) \ge \lceil (n-\ell +2 - 2k)/3 \rceil $$ γ ( G ) ≥ ⌈ ( n - ℓ + 2 - 2 k ) / 3 ⌉ . We also characterize the graphs G achieving equality for this new bound.

Keywords: Domination number; Lower bounds; Cycles; 05C69 (search for similar items in EconPapers)
Date: 2019
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://link.springer.com/10.1007/s10878-019-00409-x 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:38:y:2019:i:3:d:10.1007_s10878-019-00409-x

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

DOI: 10.1007/s10878-019-00409-x

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 ().