Labelled directed graphs, their underlying simple graphs, and label-preserving isomorphisms

Let $L$ be a set of labels. A labelled directed graph with label set $L$ is a pair $\mathrm{\Gamma }=(V,A)$ consisting of a vertex set $V$ and a set of labelled arcs

$A\subseteq V\times L\times V\mathrm{.}$

An element $(u,\lambda ,v)\in A$ is an arc from $u$ to $v$ labelled by $\lambda$. Distinct labels may connect the same ordered pair of vertices; this is why a labelled directed graph is treated here as a digraph variant rather than as a simple graph.

The underlying simple graph of $\mathrm{\Gamma }$ is the simple graph on $V$ whose edge set consists of the unordered pairs $\{u,v\}$ for which some labelled arc $(u,\lambda ,v)$ or $(v,\lambda ,u)$ exists, with loops $\{u,u\}$ discarded.

If $\mathrm{\Gamma }=(V,A)$ and ${\mathrm{\Gamma }}^{\mathrm{\prime }}=(V^{\mathrm{\prime }},A^{\mathrm{\prime }})$ are labelled directed graphs with the same label set $L$, a label-preserving directed graph isomorphism is a bijection $\phi :V\to V^{\mathrm{\prime }}$ such that

$(u,\lambda ,v)\in A⟺(\phi (u),\lambda ,\phi (v))\in A^{\mathrm{\prime }}$

for all $u,v\in V$ and $\lambda \in L$.