Total Roman domination edge-critical graphs

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.