# The identifying code number and Mycielski's construction of graphs

Document Type : Research Paper

Authors

Department of Mathematics, Imam Khomeini International University of Qazvin, P.O.Box 3414896818, Qazvin, Iran

Abstract

Let $G=(V, E)$ be a simple graph. A set $C$ of vertices $G$ is an identifying code of $G$ if for every two vertices $x$ and $y$ the sets $N_{G} [x] \cap C$ and $N_{G} [y] \cap C$ are non-empty and different. Given a graph $G,$ the smallest size of an identifying code of $G$ is called the identifying code number of $G$ and denoted by $\gamma^{ID}(G).$ Two vertices $x$ and $y$ are twins when $N_{G}[x]=N_{G}[y].$ Graphs with at least two twin vertices are not an identifiable graph. In this paper, we deal with the identifying code number of Mycielski's construction of graph $G.$ We prove that the Mycielski's construction of every graph $G$ of order $n \geq 2,$ is an identifiable graph. Also, we present two upper bounds for the identifying code number of Mycielski's construction $G,$ such that these two bounds are sharp. Finally, we show that Foucaud et al.'s conjecture is holding for Mycielski's construction of some graphs.

Keywords

Main Subjects

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### History

• Receive Date: 07 December 2020
• Revise Date: 09 August 2021
• Accept Date: 08 November 2021
• Published Online: 01 December 2022