#### Vol. 12, No. 8, 2019

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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
##### Abstract

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

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##### Keywords
Roman domination, total Roman domination, total Roman domination edge-critical graphs
Primary: 05C69
##### Milestones
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