Watching systems of triangular graphs

Document Type: Research Paper

Authors

1 Science and Research Branch, Islamic Azad University

2 Shahid Rajaee Teacher Training University

Abstract

A watching system in a graph $G=(V‎, ‎E)$ is a set $W=\{\omega_{1}‎, ‎\omega_{2}‎, ‎\dots‎, ‎\omega_{k}\}$‎, ‎where $\omega_{i}=(v_{i}‎, ‎Z_{i})‎, ‎v_{i}\in V$ and $Z_{i}$ is a subset of closed neighborhood of‎ ‎$v_{i}$ such that the sets $L_{W}(v)=\{\omega_{i}‎: ‎v\in‎ ‎Z_{i}\}$ are non-empty and distinct‎, ‎for any $v\in V$‎. ‎In this‎ ‎paper‎, ‎we study the watching systems of line graph $K_{n}$ which is‎ ‎called triangular graph and denoted by $T(n)$‎. ‎The minimum size of a‎ ‎watching system of $G$ is denoted by $\omega(G)$‎. ‎We show that‎ ‎$\omega(T(n))=\lceil\frac{2n}{3}\rceil$‎.

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