Doost Ali Mojdeh

Given a simple graph $G=(V,E)$ with maximum degree $\Delta$. Let $(V_{0},V_{1},V_{2})$ be an ordered partition of $V$, where $V_{i}=\{v\in V:f(v)=i\}$ for $i=0,1$ and $V_{2}=\{v\in V:f(v)\geq2\}$. A function $f:V\rightarrow\{0,1,...,\lceil\frac{\Delta}{2}\rceil+1\}$ is a strong Roman dominating function (StRDF) on $G$, if every $v\in V_{0}$ has a neighbor $w\in V_{2}$ and $f(w)\geq1+\lceil\frac{1}{2}|N(w)\cap V_{0}|\rceil $. A function $f:V\rightarrow\{0,1,...,\lceil\frac{\Delta}{2}\rceil+1\}$ is a unique response strong Roman function (URStRF), if $w\in V_{0}$, then $|N(w)\cap V_{2}|\leq1$ and $w\in V_{1}\cup V_{2}$ implies that $|N(w)\cap V_{2}|=0$. A function $f:V\rightarrow\{0,1,...,\lceil\frac{\Delta}{2}\rceil+1\}$ is a unique response strong Roman dominating function (URStRDF) if it is both URStRF and StRDF. The unique response strong Roman domination number of $G$, denoted by $u_{StR}% (G)$, is the minimum weight of a unique response strong Roman dominating function. In this paper we approach the problem of a Roman domination-type defensive strategy under multiple simultaneous attacks and begin with the study of several mathematical properties of this invariant. We obtain several bounds on such a parameter and give some realizability results for it. Moreover, for any tree $T$ of order $n\ge 3$ we prove the sharp bound $u_{StR}(T)\leq\frac {8n}{9}$.