Distance-Dominating Cycles in $P_3$-Dominated Graphs

Abstract

Let $G$ be a connected graph. For $x,y\in V(G)$ with $d(x,y)=2$, we define $J(x,y)=\{u\in N(x)\cap N(y)|N[u]\cap N[x]\cup N[y]\}$ and $J^{\prime }(x,y)=\{u\in N(x)\cap N(y)|$ if $v\in N(u)\setminus (N[x]\cup N[y])$ then $N(x)\cup N(y)\cup N(u)\cap N[v]\}$. A graph $G$ is quasi-claw-free if $J(x,y)\ne \mathrm{\varnothing }$ for each pair $(x,y)$ of vertices at distance $2$ in $G$. Broersma and Vumar introduced the class of $P_3$-dominated graphs defined as $J(x,y)\cup J^{\prime }(x,y)\ne \mathrm{\varnothing }$ for each $x,y\in V(G)$ with $d(x,y)=2$. Let $\kappa (G)$ and ${\alpha }_2(G)$ be the connectivity of $G$ and the maximum number of vertices that are pairwise at distance at least $2$ in $G$, respectively. A cycle $C$ is $m$-dominating if $d(x,C)=min\{d(x,u)|u\in V(C)\}\le m$ for all $x\in V(G)$. In this note, we prove that every $2$-connected ${\mathcal{P}}_3$-dominated graph $G$ has an $m$-dominating cycle if ${\alpha }_{2m+3}(G)\le \kappa (G)$.