Prove that \strongly connected" is an equivalence relation on the vertex set of a directed graph

Expert Solution

Definition:

We say that a vertex a is strongly connected to b if there exist two paths, one from a to b and another from b to a.

Note that we allow the two paths to share vertices or even to share edges. We will use a ~ b as shorthand for "a is strongly connected to b". We will allow very short paths, with one vertex and no edges, so that any vertex is strongly connected to itself.

Proof :

An equivalence relation a # b is a relation that satisfies three simple properties:

Reflexive property: For all a, a # a. Any vertex is strongly connected to itself, by definition.
Symmetric property: If a # b, then b # a. For strong connectivity, this follows from the symmetry of the definition. The same two paths (one from a to b and another from b to a) that show that a ~ b, looked at in the other order (one from b to a and another from a to b) show that b ~ a.
Transitive property: If a # b and b # c, then a # c. Let's expand this out for strong connectivity: if a ~ b and b ~ c, we have four paths: a-b, b-a, b-c, and c-b. Concatenating them in pairs a-b-c and c-b-a produces two paths connecting a-c and c-a, so a ~ c, showing that the transitive property holds for strong connectivity.

Since all three properties are true of strong connectivity, strong connectivity is an equivalence relation.

Colby Messinger answered 2 years ago

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