The identification numbers of lollipop graphs

A nontrivial connected graph $G$ with diameter $d$ can be assigned a red-white coloring, where the vertices of $G$ are colored either red or white, with the stipulation that at least one vertex must be red. Associated with each vertex $v$ of $G$ is a $d$-vector, called the code of $v$, whose $i$th coordinate is the number of red vertices at distance $i$ from $v$. A red-white coloring of $G$ for which distinct vertices have distinct codes is called an identification coloring or $ID$-coloring of $G$. A graph $G$ possessing an $ID$-coloring is called an $ID$-graph. The minimum number of red vertices among all $ID$-colorings of an $ID$-graph $G$ is the identification number or $ID$-number of $G$. The number of red vertices in an identification coloring is called the identification coloring number. This article studied the identification coloring number of lollipop graphs by constructing vertex colorings.