Total Roman domination edge-critical graphs

Lampman, Chloe; Mynhardt, Kieka (C. M.); Ogden, Shannon

doi:10.2140/involve.2019.12.1423

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Total Roman domination edge-critical graphs

Chloe Lampman, Kieka (C. M.) Mynhardt and Shannon Ogden

Vol. 12 (2019), No. 8, 1423–1439

DOI: 10.2140/involve.2019.12.1423

Abstract

A total Roman dominating function on a graph $G$ is a function $f : V (G) \to {0, 1, 2}$ such that every vertex $v$ with $f (v) = 0$ is adjacent to some vertex $u$ with $f (u) = 2$ , and the subgraph of $G$ induced by the set of all vertices $w$ such that $f (w) > 0$ has no isolated vertices. The weight of $f$ is $\sum_{v \in V (G)} f (v)$ . The total Roman domination number $γ_{t R} (G)$ is the minimum weight of a total Roman dominating function on $G$ . A graph $G$ is $k$ - $γ_{t R}$ -edge-critical if $γ_{t R} (G + e) < γ_{t R} (G) = k$ for every edge $e \in E (\bar{G}) \neq \emptyset$ , and $k$ - $γ_{t R}$ -edge-supercritical if it is $k$ - $γ_{t R}$ -edge-critical and $γ_{t R} (G + e) = γ_{t R} (G) - 2$ for every edge $e \in E (\bar{G}) \neq \emptyset$ . We present some basic results on $γ_{t R}$ -edge-critical graphs and characterize certain classes of $γ_{t R}$ -edge-critical graphs. In addition, we show that, when $k$ is small, there is a connection between $k$ - $γ_{t R}$ -edge-critical graphs and graphs which are critical with respect to the domination and total domination numbers.

Keywords

Roman domination, total Roman domination, total Roman domination edge-critical graphs

Mathematical Subject Classification 2010

Primary: 05C69

Milestones

Received: 23 July 2019

Accepted: 26 September 2019

Published: 25 October 2019

Communicated by Anant Godbole

Authors

Chloe Lampman
Department of Mathematics and Statistics University of Victoria Victoria, BC Canada

Kieka (C. M.) Mynhardt
Department of Mathematics and Statistics University of Victoria Victoria, BC Canada

Shannon Ogden
Department of Mathematics and Statistics University of Victoria Victoria, BC Canada

Vol. 12, No. 8, 2019