Total Roman Domination Edge-Critical Graphs

Abstract

A total Roman dominating function on a graph $G$ is a function 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 ${\Sigma }_{v\in V(G)}f(v)$. The total Roman domination number ${\gamma }_{tR}(G)$ is the minimum weight of a total Roman dominating function on $G$. A graph $G$ is $k$-${\gamma }_{tR}$-edge-critical if ${\gamma }_{tR}(G+e)<{\gamma }_{tR}(G)=k$ for every edge $e\in E(\overline{G})\ne \varnothing$, and $k$-${\gamma }_{tR}$-edge-supercritical if it is $k$-${\gamma }_{tR}$-edge-critical and ${\gamma }_{tR}(G+e)={\gamma }_{tR}(G)-2$ for every edge $e\in E(\overline{G})\ne \varnothing$. 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.

15 pages, 2 figures