University of Isfahan Transactions on Combinatorics 2251-8657 11 4 2022 12 01 The identifying code number and Mycielski's construction of graphs 309 316 26088 10.22108/toc.2021.126368.1794 EN Athena Shaminejad Department of Mathematics, Imam Khomeini International University of Qazvin, P.O.Box 3414896818, Qazvin, Iran Ebrahim Vatandoost Department of Mathematics, Imam Khomeini International University of Qazvin, P.O.Box 3414896818, Qazvin, Iran Kamran Mirasheh Department of Mathematics, Imam Khomeini International University of Qazvin, P.O.Box 3414896818, Qazvin, Iran Journal Article 2020 12 07 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. https://toc.ui.ac.ir/article_26088_858b032cc710d026f084d95cbe679621.pdf