Non-existence of forbidden subgraph characterization of $H$-line graphs

$H$-line graph, denoted by $HL(G)$, is a generalization of line graph. Let $G$ and $H$ be two graphs such that $H$ has at least 3 vertices and is connected. The $H$-line graph of $G$, denoted by $HL(G)$, is that graph whose vertices are the edges of $G$ and two vertices of $HL(G)$ are adjacent if they are adjacent in $G$ and lie in a common copy of $H$. In this paper, we show that $H$-line graphs do not admit a forbidden subgraph characterization.