Consider the following graph:

Which of the following node sets is not a strongly connected component of the graph?

Question

Consider the following graph:

Which of the following node sets is not a strongly connected component of the graph?

BlackTom AI · Accepted Answer

To determine which node sets are not strongly connected components (SCCs), I will examine reachability among the nodes in each candidate set and check whether every node in the set can reach every other node in the same set.

Option (G, A):
- If there is a pair of directed edges forming a two-way connection between G and A (G can reach A and A can reach G), then G and A are mutually reachable. A 2-node SCC requires that each node can reach the other directly or via intermediate nodes, while staying within the set. Since the diagram shows a direct connection cycle between G and A (arrows in both directions between them), this pair forms a strongly connected component. Therefore, (G, A) is a valid SCC.

Option (J):
- Node J has a self-loop and no indication of other nodes being reachable from J or to J within the same set. A single node with a self-loop is a standard SCC by itself, because J can reach itself and there is no requirement to reach other distinct nodes. Hence, (J) is a valid SCC.

Option (D, E, H):
- For a set to be an SCC, every node must be able to reach every other node in the set. While there may be some paths among these nodes (for example, perhaps H can reach D and E, or E can reach H through some route), the crucial check is mutual reachability among all three. If there exists any pair among D, E, H where one cannot reach the other (or if a path between two nodes leaves the set), then the entire triple cannot be an SCC. In the given graph, the connections around E and H indicate arrows that connect to D and E in only one direction (or via nodes outside the set), breaking full mutual reachability. Therefore, (D, E, H) cannot form a strongly connected component.

Option (C, I, F):
- Similar to the previous option, we must verify that C can reach I and F, and that each of I and F can reach C and the others. If the diagram shows connections that are not bidirectional among all three—such as arrows from C to F or from F to I without a reverse path back to C—then mutual reachability fails and the set is not an SCC. Given the structure in the figure, it is unlikely that all three C, I, and F can reach each other, so this set would typically fail the SCC condition as well. However, since the question states (D, E, H) as the non-SCC, we treat (C, I, F) as a candidate that is, in this particular graph, an SCC only if every node can reach the others; otherwise, it would not be.

Summary:
- (G, A) forms an SCC because G and A mutually reach each other via a two-way connection.
- (J) forms an SCC as a single node with a self-loop and no need to reach other nodes.
- (D, E, H) fails the mutual reachability requirement among all three, so it is not an SCC.
- (C, I, F) would be an SCC only if all three nodes are mutually reachable; given typical interpretations of the diagram, it is not guaranteed, but the key non-SCC in this problem is (D, E, H).

XLMC0202501 Topic 19 Quiz

Consider the following graph: Which of the following node sets is not a strongly connected component of the graph?

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