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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References

D. Auger (2010). Minimal identifying codes in trees and planar graphs with large girth. European J. Combin.. 31 (5), 1372-1384
D. Auger, I. Charon, O. Hurdy and A. Lobstein (2013). Watching systems in graphs: an extension of identifying codes. Discrete Appl. Math.. 161 (12), 1674-1685
I. Charon, O. Hudry and A. Lobstein (2003). Minimizing the size of an identifying or locating-dominating code in a graph is NP-hard. Theoret. Comput. Sci.. 290 (3), 2109-2120
I. Charon, O. Hudry and A. Lobstein (2007). Extremal cardinalities for identifying and locating-dominating codes in graphs. Discrete Math.. 307 (3-5), 356-366
R. Diestel (1997). Graph theory. Translated from the 1996 German original, Graduate Texts in Mathematics, Springer-Verlag, New York. 173
M. G. Karpovsky, K. Chakrabarty and L. B. Levitin (1998). On a new class of codes for identifying vertices in graphs. IEEE Trans. Inform. Theory. 44, 599-611
F. Foucaud, E. Guerrini, M. Kovse, R. Naserasr, A. Parreau and P. Valicov (2011). Extremal graphs for the identifying code problem. European J. Combin.. 32 (4), 628-638
F. Foucauda, S. Gravierb, R. Naserasra, A. Parreaub and P. Valicova (2013). Identifying codes in line graphs. J. Graph Theory. 37 (4), 425-448
F. Foucaud, R. Klasing, A. Kosowski and A. Raspaud (2012). On the size of identifying codes in triangle-free graphs. Discrete Appl. Math.. 160 (10-11), 1532-1546