A graph is connected if and only if the number of vertices in the breadth-first traversal is the same as the number of vertices in the graph regardless of the starting vertex.

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A graph is considered connected if there is a path between every pair of vertices. In a breadth-first traversal (BFT) starting from any vertex, if all vertices are reached (i.e., the number of vertices in the BFT is the same as the number of vertices in the graph), it indicates that there is a path from the starting vertex to every other vertex, hence the graph is connected. Conversely, if a graph is connected, a BFT from any vertex will indeed visit all vertices.

A Graph Is Connected If and Only If the Number