Total Modern Roman Dominating Functions in Graphs

Abstract

Let $G=(V(G), E(G))$ be any connected graph. A function $f:V(G)\to \{0,1,2,3\}$ is a modern Roman dominating  function of $G$ if 
for each $v\in V(G)$ with $f(v)=0$, there exist $u,w \in N_G (v)$ such that $f(u)=2$ and $f(w)=3$; and
for each $v\in V(G)$ with $f(v)=1$, there exists $u \in N_G (v)$ such that $f(u)=2$ or $f(u)=3$. In addition, if every subgraph induced by the set $\{v\in V(G):f(v)>0\}$ is an isolated free, then we say that $f$ is a total modern Roman dominating function of $G.$ The minimum weight $\omega_G^{tmR}(f)=\sum_{v\in V(G)}f(v)$ of a total modern Roman dominating function $f$ of $G$ is called the total modern Roman domination number $\gamma_{tmR}(G)$ of $G$. In this paper, we initiate the study of total modern Roman domination. We characterize graphs with smaller total modern Roman domination number  and obtain the $\gamma_{tmR}(G)$ of some special graphs. Moreover, we investigate and characterize the total modern Roman domination of  the join and corona of graphs.