June 10, 2023 # Doost Ali Mojdeh

Degree: Professor
Address: Department of Mathematics, University of Mazandaran, Babolsar, Iran
Education: Ph.D in Mathematics (Graph Theory and Combinatorics)
Phone: 011-35302448
Faculty: Faculty of Mathematical Sciences

## Research

 Title Unique response strong Roman dominating functions of graphs Type Article Keywords strong Roman dominating function, unique response strong Roman (dominating) function Journal Electronic Journal of Graph Theory and Applications DOI DOI: 10.5614/ejgta.2021.9.2.18 Researchers a { color: #4f98b0; } a:hover { color: #ffab00; } a:link:visited { text-decoration: none; } Doost Ali Mojdeh (First researcher) , Guoliang Hao (Second researcher) , Iman Masoumi (Third researcher) , Ali Parsian (Fourth researcher)

## Abstract

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 $u_{StR}(T)\leq\frac {8n}{9}$.