Multigraphs, loops and directed graphs as variants distinct from the default finite simple graph

A finite undirected multigraph consists of a finite vertex set $V$, a finite edge set $M$, and an endpoint map assigning to each $e\in M$ either a singleton $\{v\}$ or a two-element subset $\{u,v\}$ of $V$. An edge assigned $\{v\}$ is a loop at $v$. Distinct members of $M$ assigned the same endpoint set are parallel edges.

A finite directed graph, or digraph, consists of a finite vertex set $V$ and a set $A\subseteq V\times V$ of arcs. An arc $(u,v)$ is directed from $u$ to $v$; an arc $(v,v)$ is a directed loop.

These objects are variants, not instances of the default graph convention in A finite simple graph is a finite vertex set together with a set of two-element vertex subsets and Unless stated otherwise, graph means finite, simple and undirected; orders, sizes and empty-set conventions are fixed here. A theorem stated merely for a graph therefore neither permits loops or parallel edges nor supplies directions to its edges.