Let $G=(V, E)$ be a simple graph. A set $C$ of vertices $G$ is an Identifying set of $G$ if for every two vertices $x$ and $y$ belong to $V$ 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 set of $G$ is called the Identifying code number of $G$ and is 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 identifiable graphs. In this paper,  we present three bounds for Identifying code number.

‎Let $G=(V‎, ‎E)$ be a simple graph‎. ‎A set $C$ of vertices of $G$ 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 $\gamma^{ID}(G).$ In this paper‎, ‎we prove that the Identifying code number of the subdivision of a graph $G$ of order $n$ is at most $n$‎. ‎Also‎, ‎we prove that the Identifying code number of the subdivision of graphs $K_n$, $K_{r,s}$ and $C_P(s)$ are $n‎$,‎ ‎‎‎$‎‎r+s$ and $2s$, respectively‎. ‎Finally‎, ‎we conjecture that for every graph $G$ of order $n$ the Identifying code number of the subdivision of $G$ is $n$‎.

‎Let $G=(V‎, ‎E)$ be a simple graph‎. ‎A set $C$ of vertices of $G$ 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 $\gamma^{ID}(G).$ In this paper‎, ‎we prove that the Identifying code number of the subdivision of a graph $G$ of order $n$ is at most $n$‎. ‎Also‎, ‎we prove that the Identifying code number of the subdivision of graphs $K_n$, $K_{r,s}$ and $C_P(s)$ are $n‎$,‎ ‎‎‎$‎‎r+s$ and $2s$, respectively‎. ‎Finally‎, ‎we conjecture that for every graph $G$ of order $n$ the Identifying code number of the subdivision of $G$ is $n$‎.

A watching system in a graph G=(V, E) is a set W={w1, w2, …,wk}, where wi=(vi, Zi); viÎ V and Zi is a subset of closed neighborhood of vi such that the sets LW(v) ={wi: vÎZi} are non-empty and distinct, for any v ÎV. In this paper, we study the watching systems of line graph Kn which is called triangular graph and denoted by T (n). The minimum size of a watching system of G is denoted by w (G). We show that w (T(n))= [2n/3].

Let $G=(V‎, ‎E)$ be a simple and undirected graph‎. ‎A watcher $\omega_i$ of $G$ is a couple of $\omega_i=(v_i‎, ‎Z_i),$ where $v_i \in V$ and $Z_i$ is a subset of the closed neighborhood of $v_i.$ If a vertex $v \in Z_i,$ we say that $v$ is covered by $\omega_i.$ A set $W=\{\omega_1‎, ‎\omega_2‎, ‎\dots‎, ‎\omega_k\}$‎, ‎of watchers is a watching system for $G$ if the sets $L_W(v)=\{\omega_i~:~v \in Z_i‎ ~,~ ‎1 \le i \le k\}$ are non-empty and distinct‎, ‎for every $v \in V$‎. ‎In this paper‎, ‎we study the watching systems of some graphs‎, ‎and consider the watching number of Mycielski's construction of some graphs‎.