Original graphs of link graphs

Let $\ell \geqslant 0$ be an integer, and $G$ be a graph without loops. An $\ell$-link of $G$ is a walk of length $\ell$ in which consecutive edges are different. We identify an $\ell$-link with its reverse sequence. The $\ell$-link graph $\mathbb{L}_\ell(G)$ of $G$ is defined to have vertices the $\ell$-links of $G$, such that two vertices of $\mathbb{L}_\ell(G)$ are adjacent if their corresponding $\ell$-links are the initial and final subsequences of an $(\ell + 1)$-link of $G$. A graph $G$ is called an $\ell$-root of a graph $H$ if $\mathbb{L}_\ell(G) \cong H$. For example, $\mathbb{L}_0(G) \cong G$. And the $1$-link graph of a simple graph is the line graph of that graph. Moreover, let $H$ be a finite connected simple graph. Whitney's isomorphism theorem (1932) states if $H$ has two connected nonnull simple $1$-roots, then $H \cong K_3$, and the two $1$-roots are isomorphic to $K_3$ and $K_{1, 3}$ respectively. This paper investigates the $\ell$-roots of finite graphs. We show that every $\ell$-root is a certain combination of a finite minimal $\ell$-root and trees of bounded diameter. This transfers the study of $\ell$-roots into that of finite minimal $\ell$-roots. As a qualitative generalisation of Whitney's isomorphism theorem, we bound from above the number, size, order and maximum degree of minimal $\ell$-roots of a finite graph. This work forms the basis for solving the recognition and determination problems for $\ell$-link graphs in our future papers. As a byproduct, we characterise the $\ell$-roots of some special graphs including cycles. Similar results are obtained for path graphs introduced by Broersma and Hoede (1989). $G$ is an $\ell$-path root of a graph $H$ if $H$ is isomorphic to the $\ell$-path graph of $G$. We bound from above the number, size and order of minimal $\ell$-path roots of a finite graph.