Total Roman Domination Edge-Supercritical and Edge-Removal-Supercritical Graphs

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 ${\mathrm{\Sigma }}_{v\in V(G)}f(v)$. 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})\neq\emptyset $,and$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 $e\in E(\bar{G})\mathrm{\ne }\mathrm{\varnothing }$. A graph $G$ is $k$-${\gamma }_{tR}$-edge-stable if ${\gamma }_{tR}(G+e)={\gamma }_{tR}(G)=k$ for every edge $e\in E(\bar{G})$ or $E(\bar{G})=\mathrm{\varnothing }$. For an edge $e\in E(G)$incidentwithadegree$1$vertex,wedefine$\gamma_{tR}(G-e)=\infty$. A graph $G$ is $k$-${\gamma }_{tR}$-edge-removal-critical if ${\gamma }_{tR}(G-e)>{\gamma }_{tR}(G)=k$ for every edge $e\in E(G)$, and $k$-${\gamma }_{tR}$-edge-removal-supercritical if it is $k$-${\gamma }_{tR}$-edge-removal-critical and ${\gamma }_{tR}(G-e)\ge {\gamma }_{tR}(G)+2$ for every edge $e\in E(G)$. A graph $G$ is $k$-${\gamma }_{tR}$-edge-removal-stable if ${\gamma }_{tR}(G-e)={\gamma }_{tR}(G)=k$ for every edge $e\in E(G)$. We investigate connected ${\gamma }_{tR}$-edge-supercritical graphs and exhibit infinite classes of such graphs. In addition, we characterize ${\gamma }_{tR}$-edge-removal-critical and ${\gamma }_{tR}$-edge-removal-supercritical graphs. Furthermore, we present a connection between $k$-${\gamma }_{tR}$-edge-removal-supercritical and $k$-${\gamma }_{tR}$-edge-stable graphs, and similarly between $k$-${\gamma }_{tR}$-edge-supercritical and $k$-${\gamma }_{tR}$-edge-removal-stable graphs.