On the watching number of graphs

Document Type : Research Article


Department of‎ ‎pure Mathematics‎, ‎Faculty of science, Imam Khomeini International University‎, ‎P.O.Box 34149-16818, Qazvin‎, ‎Iran.


Let $G=(V‎, ‎E)$ be a simple and undirected graph‎. ‎A watcher $\omega_i$ of $G$ is a couple of $\omega_i=(v_i‎, ‎Z_i),$ where $v_i \in V$ and $Z_i$ is a subset of the closed neighborhood of $v_i.$ If a vertex $v \in Z_i,$ we say that $v$ is covered by $\omega_i.$ A set $W=\{\omega_1‎, ‎\omega_2‎, ‎\dots‎, ‎\omega_k\}$‎, ‎of watchers is a watching system for $G$ if the sets $L_W(v)=\{\omega_i~:~v \in Z_i‎ ~,~ ‎1 \le i \le k\}$ are non-empty and distinct‎, ‎for every $v \in V$‎. ‎In this paper‎, ‎we study the watching systems of some graphs‎, ‎and consider the watching number of Mycielski's construction of some graphs‎.


Main Subjects

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