Signed total Roman domination and domatic numbers in graphs

Abstract

A signed total Roman dominating function (STRDF) on a graph G is a function $f:V\left(G\right)⟶\{-1,1,2\}$ satisfying (i) ${\sum }_{x\in N_G\left(u\right)}f\left(x\right)\ge 1$ for each vertex $u\in V\left(G\right)$ and its neighborhood $N_G\left(u\right)$ in G and, (ii) every vertex $u\in V\left(G\right)$ with $f\left(u\right)=-1$, there exists a vertex $v\in N_G\left(u\right)$ with $f\left(v\right)=2$. The minimum number ${\sum }_{u\in V\left(G\right)}f\left(u\right)$ among all STRDFs f on G is denoted by ${\gamma }_{stR}\left(G\right)$. A set $\{f_1,\dots ,f_d\}$ of distinct STRDFs on G is called a signed total Roman dominating family on G if ${\sum }_{i=1}^d{f}_i\left(u\right)\le 1$ for each $u\in V\left(G\right)$. We use $d_{stR}\left(G\right)$ to denote the maximum number of functions among all signed total Roman dominating families on G. Our purpose in this paper is to examine the effects on ${\gamma }_{stR}\left(G\right)$ when G is modified by removing or subdividing an edge. In addition, we determine the number $d_{stR}\left(G\right)$ for the case that G is a complete graph or bipartite graph.