Nordhaus–Gaddum bounds for total Roman domination

J. Amjadi (), S. M. Sheikholeslami () and M. Soroudi ()
Additional contact information
J. Amjadi: Azarbaijan Shahid Madani University
S. M. Sheikholeslami: Azarbaijan Shahid Madani University
M. Soroudi: Azarbaijan Shahid Madani University

Journal of Combinatorial Optimization, 2018, vol. 35, issue 1, No 10, 126-133

Abstract: Abstract A Nordhaus–Gaddum-type result is a lower or an upper bound on the sum or the product of a parameter of a graph and its complement. In this paper we continue the study of Nordhaus–Gaddum bounds for the total Roman domination number $$\gamma _{tR}$$ γ t R . Let G be a graph on n vertices and let $$\overline{G}$$ G ¯ denote the complement of G, and let $$\delta ^*(G)$$ δ ∗ ( G ) denote the minimum degree among all vertices in G and $$\overline{G}$$ G ¯ . For $$\delta ^*(G)\ge 1$$ δ ∗ ( G ) ≥ 1 , we show that (i) if G and $$\overline{G}$$ G ¯ are connected, then $$(\gamma _{tR}(G)-4)(\gamma _{tR}(\overline{G})-4)\le 4\delta ^*(G)-4$$ ( γ t R ( G ) - 4 ) ( γ t R ( G ¯ ) - 4 ) ≤ 4 δ ∗ ( G ) - 4 , (ii) if $$\gamma _{tR}(G), \gamma _{tR}(\overline{G})\ge 8$$ γ t R ( G ) , γ t R ( G ¯ ) ≥ 8 , then $$\gamma _{tR}(G)+\gamma _{tR}(\overline{G})\le 2\delta ^*(G)+5$$ γ t R ( G ) + γ t R ( G ¯ ) ≤ 2 δ ∗ ( G ) + 5 and (iii) $$\gamma _{tR}(G)+\gamma _{tR}(\overline{G})\le n+5$$ γ t R ( G ) + γ t R ( G ¯ ) ≤ n + 5 and $$\gamma _{tR}(G)\gamma _{tR}(\overline{G})\le 6n-5$$ γ t R ( G ) γ t R ( G ¯ ) ≤ 6 n - 5 .

Keywords: Total Roman dominating function; Total Roman domination number; Nordhaus–Gaddum inequalities (search for similar items in EconPapers)
Date: 2018
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://link.springer.com/10.1007/s10878-017-0158-5 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:35:y:2018:i:1:d:10.1007_s10878-017-0158-5

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

DOI: 10.1007/s10878-017-0158-5

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 ().