# Pairs of paths with the same source and target

Commutative diagrams usually express path equivalences in a category and thus involve pairs of paths in a category with the same source and target.

General diagrams - in categories resp. category theory - do not necessarily involve pairs of paths with the same source and target.

But is there a name for pairs of paths with the same source and target in quivers?
Pairs which only eventually are to be called "commutative" (expressing equivalence)?
("Diagram" would be a badly chosen name.)

The only commutative closed shapes that are explicitely named are "commutative triangles" and "commutative squares". So I thought about "commutative polygons" but a Google search for "commutative polygons" gave only 10 hits.

Furthermore the question is actually only about a very restricted kind of polygons, maybe "bi-directed" polygons?

If there is no established name, I would appreciate any suggestion.

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Pairs of paths with the same source and target but with no other nodes in common are called parallel paths, at least on the computer science side of things in graph theory -- you can google the term to get some relevant CS papers, but I couldn't find a wikipedia article on parallel paths.

If your paths are allowed to share nodes other than the source and target, I would suggest something like "piecewise parallel paths". With this terminology you can simultaneously get the idea across and score cheap points for alliteration.

As requested in the comments, here is a reference

Jwo et. al., Characterization of node disjoint (parallel) path in star graphs (1991) see here.

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I would have appreciated a singular term - like "diagram" and not mentioning "paths" - but "parallel paths" sounds perfectly right. (I will not make the distinction between piecewise or not.) – Hans Stricker Aug 4 '13 at 23:58
A reference to at least one relevant article about "parallel paths" would be appreciated a lot! – Hans Stricker Aug 5 '13 at 0:08
If someone called two paths piecewise parallel, I'd probably assume that the common vertices (if any) occur in the same order along both paths. – Andreas Blass Aug 5 '13 at 1:43
a newer reference is Bhandari, Ramesh (1999). Survivable networks: algorithms for diverse routing 477. Springer. p. 46. ISBN 0-7923-8381-8. Efficient algorithms were initially developed by Suurballe and Tarjan. – Manfred Weis Aug 5 '13 at 10:25