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Precisely speaking, what is the difference between the graph terms of ("vertex" vs. "node") and ("edge" vs. "arc")?

I have read that "node" and "arc" should be used when the graph is strictly a tree.

If there is a precise rule or protocol, please cite a reference.

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As far as I can tell, there is no difference. Some people use some terms, and some people prefer others. Perhaps some textbooks make a difference (one idea would be to use different words for directed and undirected graphs), but I'm not aware of any, and these distinctions are not standard anyway. –  Yuval Filmus Apr 5 '11 at 23:53
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up vote 8 down vote accepted

The distinction between vertex and node seems to me to be mostly about discipline (e.g. whether you come from combinatorics or computer science) and is irrelevant. The distinction between edge and arc can sometimes be relevant depending on who's using it: combinatorialists sometimes use "edge" to mean "undirected edge" and "arc" to mean "directed edge," although this usage is not universal.

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In addition, "arc" is often used in topological situations; e.g., Diestel defines a polygonal arc as a union of finitely many straight line segments homeomorphic to [0,1] when talking about planar graphs. –  Harry Stern Apr 6 '11 at 0:19
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My readings suggests "arc" and "edge" are conceptually the same. Yet, LEMON ( http://lemon.cs.elte.hu/trac/lemon )has separate functions/methods for arcs & edges. I have played with it but not enough to understand the difference, in their usage.

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