**Authors:** E. Shabani, N. Jafari Rad, A. Poureidi

**Keywords:** Domination, Roman domination, Hop Roman domination.

### Abstract

A subset $S$ of vertices of a graph $G$ is a hop dominating set
if every vertex outside $S$ is at distance two from a vertex of
$S$. A Roman dominating function on a graph $G=(V,E)$ is a
function $f: V(G) \longrightarrow \{0, 1, 2\}$ satisfying the
condition that every vertex $u$ for which $f(u)=0$ is adjacent to
at least one vertex $v$ for which $f(v)=2$. A hop Roman
dominating function (HRDF) of $G$ is a function $f: V(G)
\longrightarrow \{0, 1, 2\}$ having the property that for every
vertex $v \in V$ with $f(v) = 0$ there is a vertex $u$ with
$f(u)=2$ and $d(u,v)=2$. The weight of a HRDF $f$ is the sum $f
(V) = \sum_{v\in V} f(v)$. The minimum weight of a HRDF on $G$ is
called the hop Roman domination number of $G$ and is denoted by
$\gamma_{hR}(G)$. In this paper we characterize all graphs $G$ of
order $n$ with $\gamma_{hR}(G)=n$ or $\gamma_{hR}(G)=n-1$.

School of Mathematical Sciences,

Shahrood University of Technology,

Shahrood, Iran

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