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 Covering Italian domination in graphs Type Article Keywords Covering Italian domination number. Vertex cover number. Claw-free graphs Journal DISCRETE APPLIED MATHEMATICS DOI https://doi.org/10.1016/j.dam.2021.08.001 Researchers a { color: #4f98b0; } a:hover { color: #ffab00; } a:link:visited { text-decoration: none; } Abdollah Khodkar (First researcher) , Doost Ali Mojdeh (Second researcher) , Babak Samadi (Third researcher) , Ismael Gonzalez Yero (Fourth researcher)

## Abstract

For a graph $G=(V(G),E(G))$, an Italian dominating function (ID function) of $G$ is a function $f:V(G)\rightarrow \{0,1,2\}$ such that for each vertex $v\in V(G)$ with $f(v)=0$, $f(N(v))\geq2$, that is, either there is a vertex $u \in N(v)$ with $f (u) = 2$ or there are two vertices $x,y\in N(v)$ with $f(x)=f(y)=1$. A function $f:V(G)\rightarrow \{0,1,2\}$ is a covering Italian dominating function (CID function) of $G$ if $f$ is an ID function and $\{v\in V(G)\mid f(v)\neq0\}$ is a vertex cover set. The covering Italian domination number (CID number) $\gamma_{cI}(G)$ is the minimum weight taken over all CID functions of $G$. In this paper, we study the CID number in graphs. We show that the problem of computing this parameter is NP-hard even when restricted to some well-known families of graphs, and find some bounds on this parameter. We characterize the family of graphs for which their CID numbers attain the upper bound twice their vertex cover number as well as all claw-free graphs whose CID numbers attain the lower bound half of their orders. We also give the characterizations of some families of graphs with small CID numbers.