Infinite paths. Let G = (V, E) be a directed graph with a designated “start vertex” s ϵ V, a set VG ⊆ V of “good” vertices, and a set VB ⊆ V of “bad” vertices. An infinite trace p of G is an infinite sequence v0v1v2 ⋯ of vertices vi ϵ V such that (1) v0 = s, and (2) for all i ≥ 0, (vi, vi+1) ∈ E. That is, p is an infinite path in G starting at vertex s. Since the set V of vertices is finite, every infinite trace of G must visit some vertices infinitely often.
(a) If p is an infinite trace, let Inf(p) ⊆ V be the set of vertices that occur infinitely often in p. Show that Inf(p) is a subset of a strongly connected component of G.
(b) Describe an algorithm that determines if G has an infinite trace.
(c) Describe an algorithm that determines if G has an infinite trace that visits some good vertex in VG infinitely often.
(d) Describe an algorithm that determines if G has an infinite trace that visits some good vertex in VG infinitely often, but visits no bad vertex in VB infinitely often.
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