Strong Roman Domination Number of Complementary Prism Graphs

Authors : Doost Ali MOJDEH, Ali PARSİAN, İman MASOUMİ

Pages : 40-47

View : 9 | Download : 4

Publication Date : 2019-06-30

Article Type : Research Paper

Abstract :Let $G=insert ignore into journalissuearticles values(V,E);$ be a simple graph  with vertex set $V=Vinsert ignore into journalissuearticles values(G);$, edge set $E=Einsert ignore into journalissuearticles values(G);$ and from maximum degree $\Delta=\Deltainsert ignore into journalissuearticles values(G);$. Also let $f:V\rightarrow\{0,1,...,\lceil\frac{\Delta}{2}\rceil+1\}$ be a function that labels the vertices of $G$. Let $V_i=\{v\in V: finsert ignore into journalissuearticles values(v);=i\}$ for $i=0,1$ and let $V_2=V-insert ignore into journalissuearticles values(V_0\bigcup V_1);=\{w\in V: finsert ignore into journalissuearticles values(w);\geq2\}$. A function $f$ is called a strong Roman dominating function insert ignore into journalissuearticles values(StRDF); for $G$, if every $v\in V_0$ has a neighbor $w$, such that $w\in V_2$ and $finsert ignore into journalissuearticles values(w);\geq 1+\lceil\frac{1}{2}|Ninsert ignore into journalissuearticles values(w);\bigcap V_0|\rceil$. The minimum weight, $\omegainsert ignore into journalissuearticles values(f);=finsert ignore into journalissuearticles values(V);=\Sigma_{v\in V} finsert ignore into journalissuearticles values(v);$, over all the strong Roman dominating functions of $G$, is called the strong Roman domination number of $G$ and we denote it by $\gamma_{StR}insert ignore into journalissuearticles values(G);$. An StRDF of minimum weight is called a $\gamma_{StR}insert ignore into journalissuearticles values(G);$-function. Let $\overline{G}$ be the complement of $G$. The complementary prism $G\overline{G}$ of $G$ is the graph formed from the disjoint union $G$ and $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. In this paper, we investigate some properties of Roman, double Roman and strong Roman domination number of  $G\overline{G}$.
Keywords : Strong Roman domination, double Roman domination, Roman domination, prism, complementary prism