An Algorithms Question
Define a “hub” to be a vertex that shares an edge with every other vertex (such as the middle of a star graph, or any vertex in a complete graph). Suppose we have the adjacency matrix of an undirected, unweighted graph with V vertices (so our input has size V^2). Find an algorithm with running time o(V^2) that can determine whether a hub exists in the graph, or prove that no such algorithm exists. Note the use of little-o notation: the algorithm must be asymptotically faster than (big) O(V^2).
My bet is that no such algorithm exists, but I can’t figure out how to prove it.
I’ll be at Mudd tomorrow, btw.