Proposition: If a graph is connected by walks, then it is also connected by paths. That is, suppose any two vertices x,y ∈ V can be connected by a sequence of neighboring vertices, a walk W = v(0)v(1)...v(k) where each step is along an edge, v(i)v(i+1) ∈ E for all i, and v(0) = x , v(k) = y. In W, we may come back to the same vertex several times, but the Proposition says we can cut W down to a path P of neighboring vertices with no repeats, P = v(0)v(i1)v(i2)...v(k) which skips from vertex 0 to vertex i1 to vertex i2, etc., before finishing at v(k) = y, but each step is still along an edge.

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