University of IsfahanTransactions on Combinatorics2251-865711420221201The identifying code number and Mycielski's construction of graphs3093162608810.22108/toc.2021.126368.1794ENAthenaShaminejadDepartment of Mathematics, Imam Khomeini International University of Qazvin, P.O.Box 3414896818, Qazvin, IranEbrahimVatandoostDepartment of Mathematics, Imam Khomeini International University of Qazvin, P.O.Box 3414896818, Qazvin, IranKamranMirashehDepartment of Mathematics, Imam Khomeini International University of Qazvin, P.O.Box 3414896818, Qazvin, IranJournal Article20201207Let $G=(V, E)$ 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