The identifying code number and Mycielski's construction of graphs

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Shaminejad Athena; Vatandoost Ebrahim; Mirasheh Kamran

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Journal Paper

Paper Information

Journal: TRANSACTIONS ON COMBINATORICS Year:2022 | Volume:11 | Issue:4 Page(s): 309-316

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Information Journal Paper

Title

The identifying code number and Mycielski's construction of graphs

Author(s)

Shaminejad Athena | Vatandoost Ebrahim | Mirasheh Kamran | Issue Writer Certificate

Keywords

dominating set

Identifying code

Mycielski's Construction

Identifiable Graph

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.

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