Signed Roman (Total) Domination Numbers of Complete Bipartite Graphs and Wheels

Abstract

A signed (res. signed total) Roman dominating function, SRDF (res. STRDF) for short, of a graph $G = (V, E)$ is a function $f : V$ → {$−1, 1, 2$} satisfying the conditions that (i) $\sum\limits_{v∈N[v]}f(v) ≥ 1$ (res. $\sum\limits_{v∈N[v]}f(v) ≥ 1$) for any $v ∈ V$ , where $N[v]$ is the closed neighborhood and $N(v)$ is the neighborhood of $v$, and (ii) every vertex $v$ for which $f(v) = −1$ is adjacent to a vertex $u$ for which $f(u) = 2$. The weight of a SRDF (res. STRDF) is the sum of its function values over all vertices. The signed (res. signed total) Roman domination number of $G$ is the minimum weight among all signed (res. signed total) Roman dominating functions of $G$. In this paper, we compute the exact values of the signed (res. signed total) Roman domination numbers of complete bipartite graphs and wheels.