The transitive closure of a *directed graph*, is another directed graph which encodes the reachability of nodes from other nodes. If $G$ is a graph, the edge $(v\_1,v\_2)$ is in it's transitive closure $G^{tc}$ iff there is a directed path from $v\_1$ to $v\_2$ in $G$.


A *multigraph* can have multiple edges between nodes.
The question is what would be natural definitions for the transitive closure of a multigraph?


An obvious answer would be the transitive closure of the induced graph (same graph with multiple edges between verices replaced with a single edge).


Are there already interesting graphs derivable from a multigraph which could earn the title of 'transitive closure'?



Transitive closure of multigraphs
Note that the term *transitive closure* comes from set theory. Every (simple) directed graph $G$ naturally defines a relation $R(G)$ on $V(G)$. The transitive closure $G'$ of $G$ is the (simple) directed graph $G'$ on $V(G)$ such that $R(G')$ is the transitive closure of $R(G)$. I am certainly not an expert, but I guess we need to generalize the notion of relation, where an element can be related to another element with multiplicity. 


Anyway, after all that rambling another possible answer is for each $u, v \in V(G)$, put $n(u,v)$ directed edges from $u$ to $v$, where $n(u,v)$ is the maximum number of internally disjoint directed paths from $u$ to $v$. 


