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

TL;DR

This paper studies the properties of graphs related to total Roman domination, focusing on edge-critical, supercritical, and stable graphs, and provides characterizations and infinite classes of such graphs.

Contribution

It introduces new classifications of graphs based on total Roman domination and characterizes their properties and relationships, including infinite classes.

Findings

Characterized $ ext{γ}_{tR}$-edge-removal-critical graphs.

Identified infinite classes of $ ext{γ}_{tR}$-edge-supercritical graphs.

Established connections between removal-supercritical and removal-stable graphs.

Abstract

A total Roman dominating function on a graph $G$ is a function $f:V(G)\rightarrow \{0,1,2\}$ 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 $\Sigma_{v\in V(G)}f(v)$. 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(\overline{G})\neq \emptyset $. A graph $G$ is $k$-$\gamma_{tR}$-edge-stable if $\gamma_{tR}(G+e)=\gamma _{tR}(G)=k$ for every edge $e\in E(\overline{G})$ or $E(\overline{G})=\emptyset$. For an edge $e\in E(G)$ incident with a degree $1$ vertex, we define $\gamma_{tR}(G-e)=\infty$. 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 $\gamma_{tR}(G-e)\geq\gamma_{tR}(G)+2$ 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.