The distinguishing number and the distinguishing index of line and graphoidal graphs

The distinguishing number (index) $D(G)$ ($D^{\mathrm{\prime }}(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. A graphoidal cover of $G$ is a collection $\psi$ of (not necessarily open) paths in $G$ such that every path in $\psi$ has at least two vertices, every vertex of $G$ is an internal vertex of at most one path in $\psi$ and every edge of $G$ is in exactly one path in $\psi$. Let $\mathrm{\Omega }(G,\psi )$ denote the intersection graph of $\psi$. A graph $H$ is called a graphoidal graph, if there exists a graph $G$ and a graphoidal cover $\psi$ of $G$ such that $H\cong \mathrm{\Omega }(G,\psi )$. In this paper, we study the distinguishing number and the distinguishing index of the line graph and the graphoidal graph of a simple connected graph $G$.