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The Reachability & SCC Correlation

In a directed graph \(G = (V, E)\), we explore the relationship between path reachability and Strongly Connected Components (SCCs). To analyze this, we define two functions:

  • \(f(u, v) \in \{True, False\}\): Returns True if there exists at least one directed path from \(u\) to \(v\).
  • \(scc\_id(u) \in \mathbb{Z}\): Returns a unique ID for the SCC containing node \(u\).

In the following example, we can observe that two nodes \(u\) and \(v\) are in the same SCC if and only if \(scc\_id(u) = scc\_id(v)\).

scc_id = 1 scc_id = 2 scc_id = 3 1 2 3 4

Consider the visualization above. In this specific case, \(scc\_id(1) = scc\_id(2)\), and we can see that \(f(1, 3)\) is True.

Problem Statement

For each of the following logical statements, determine whether it is True (T) or False (F) for any arbitrary directed graph \(G\):

  1. \(scc\_id(u) = scc\_id(v) \implies f(u, v)\)
  2. \(f(u, v)\) AND \(f(v, u) \implies scc\_id(u) = scc\_id(v)\)
  3. \(f(u, v)\) AND \(f(v, u) \iff scc\_id(u) = scc\_id(v)\)
  4. \(f(u, v) \implies scc\_id(u) = scc\_id(v)\)
  5. \(f(u, v)\) AND \(f(v, w) \implies f(u, w)\)
  6. NOT \(f(u, v)\) AND NOT \(f(v, u) \implies scc\_id(u) \neq scc\_id(v)\)

Alternatives

a) TTTFTT
b) TTTFFF
c) TFTFTT
d) TTTTTT
e) None of the above

Original idea by: Luis Alberto Vásquez Vargas

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