TY - JOUR
T1 - Signed Roman (Total) Domination Numbers of Complete Bipartite Graphs and Wheels
AU - Zhao , Yancai
AU - Miao , Lianying
JO - Communications in Mathematical Research 
VL - 4
SP - 318
EP - 326
PY - 2019
DA - 2019/11
SN - 33
DO - http://doi.org/10.13447/j.1674-5647.2017.04.04
UR - https://global-sci.org/intro/article_detail/cmr/13413.html
KW - signed Roman domination, signed total Roman domination, complete bipartite graph, wheel
AB - <p style="text-align: justify;">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.</p>