The Perfect Locating Signed Roman Domination of some Graphs


Abolape Deborah Akwu, Tayo Charles Adefokun, Opeyemi Oyewumi




In this paper, we introduce the concept of Perfect locating signed Roman dominating functions in graphs. A perfect locating signed Roman dominating $PLSRD$ function of a graph $G=(V,E)$ is a function $f:V(G)\rightarrow\{-1,1,2\}$ satisfying the conditions that for (i) every vertex $v$ with $f(v)=-1$ is adjacent to exactly one vertex $u$ with $f(u)=2$; (ii) any pair of distinct vertices $v,w$ with $f(v)=f(w)=-1$ does not have a common neighbor $u$ with $f(u)=2$ and (iii) $f(v) + \sum_{u\in N(v)} f(u)\geq 1$ for any vertex $v$. The weight of $PLSRD$- function is the sum of its function values over all the vertices. The perfect locating signed Roman domination number of $G$ denoted by $\gamma_{LSR}^P(G)$ is the minimum weight of a $PLSRD$- function in $G$. We present the upper and lower bonds of $PLSRD$- function for trees. In addition, for grid graph $G$, we show that $\gamma_{LSR}^P(G)\leq \frac{3}{4}|G|$.