论文标题
罗马统治的总统治优势和边缘侵略性图形
Total Roman Domination Edge-Supercritical and Edge-Removal-Supercritical Graphs
论文作者
论文摘要
图形$ g $上的总罗马统治功能是一个函数$ f：v（g）\ rightarrow \ {0,1,2 \} $顶点。 $ f $的重量为$σ_{v \ in V（g）} f（v）$。总罗马统治号码$γ_{tr}（g）$是$ g $的总罗马统治功能的最小重量。图$ g $是$ k $ - $γ_{tr} $ - 如果$γ_{tr}（g+e）（g+e）<γ_{tr}（g）= k $ for e（\ overline {g} {g}）中的每个边缘$ e \ in（\ neq \ neq \ emptyset $ expryset $ exper ins） $ k $ - $γ_{tr} $ - edge-Crition-Critical和$γ_{tr}（g+e）=γ_{tr}（g）-2 $ for e（\ overline {g}）\ neq \ nebyset $ in E（\ overline {g}）中的每个边缘$ e \。图$ g $是$ k $ - $γ_{tr} $ - edge-edge-gene如果$γ_{tr}（g+e）=γ_{tr}（g）= k $ for e（\ overline {g}）$ e（\ overline {g}）$ e（\ everline {g}）$ e（\ overline {g}）对于E（g）$中的边缘$ e \，具有$ 1 $顶点的事件，我们定义$γ_{tr}（g-e）= \ infty $。图$ g $是$ k $ - $γ_{tr} $ - edge-edge-removal-关键 - 如果$γ_{tr}（g-e）>γ_{tr}（g）= k $ for e（g）$ in Edge $ e \ in Edge $ e \ e（g）$，以及$ k $ - $ k $ - $umγ_{trean} $ - expe-ed} $ - expem-gremaval-premalital-efemeritaligation In是$ k $ - $γ_{tr} $ - edge-gremoval-Critial-clitical-和$γ_{tr}（g-e）\geqγ_{tr}（g）+2 $ for e（g）$中的每个边缘$ e \ for e（g）$。图$ g $是$ k $ - $γ_{tr} $ - edge-edge-removal-stable如果$γ_{tr}（g-e）=γ_{tr}（g）= k $ for e（g）$中的每个edge $ e \ for e（g）$。我们研究了连接的$γ_{tr} $ - 边缘cul危图，并展示了此类图的无限类。此外，我们表征了$γ_{tr} $ - edge-edge-removal-关键 - $γ_{tr} $ - edge-edge-removal-supergalitical thrage。 Furthermore, we present a connection between $k$-$γ_{tR}$-edge-removal-supercritical and $k$-$γ_{tR}$-edge-stable graphs, and similarly between $k$-$γ_{tR}$-edge-supercritical and $k$-$γ_{tR}$-edge-removal-stable graphs.
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 $Σ_{v\in V(G)}f(v)$. The total Roman domination number $γ_{tR}(G)$ is the minimum weight of a total Roman dominating function on $G$. A graph $G$ is $k$-$γ_{tR}$-edge-critical if $γ_{tR}(G+e)<γ_{tR}(G)=k$ for every edge $e\in E(\overline{G})\neq\emptyset $, and $k$-$γ_{tR}$-edge-supercritical if it is $k$-$γ_{tR}$-edge-critical and $γ_{tR}(G+e)=γ_{tR}(G)-2$ for every edge $e\in E(\overline{G})\neq \emptyset $. A graph $G$ is $k$-$γ_{tR}$-edge-stable if $γ_{tR}(G+e)=γ_{tR}(G)=k$ 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 $γ_{tR}(G-e)=\infty$. A graph $G$ is $k$-$γ_{tR}$-edge-removal-critical if $γ_{tR}(G-e)>γ_{tR}(G)=k$ for every edge $e\in E(G)$, and $k$-$γ_{tR}$-edge-removal-supercritical if it is $k$-$γ_{tR}$-edge-removal-critical and $γ_{tR}(G-e)\geqγ_{tR}(G)+2$ for every edge $e\in E(G)$. A graph $G$ is $k$-$γ_{tR}$-edge-removal-stable if $γ_{tR}(G-e)=γ_{tR}(G)=k$ for every edge $e\in E(G)$. We investigate connected $γ_{tR}$-edge-supercritical graphs and exhibit infinite classes of such graphs. In addition, we characterize $γ_{tR}$-edge-removal-critical and $γ_{tR}$-edge-removal-supercritical graphs. Furthermore, we present a connection between $k$-$γ_{tR}$-edge-removal-supercritical and $k$-$γ_{tR}$-edge-stable graphs, and similarly between $k$-$γ_{tR}$-edge-supercritical and $k$-$γ_{tR}$-edge-removal-stable graphs.
