Question

I have an undirected unweighted graph G = (V, E) and a randomly chosen subset S of its vertices. I want to check whether the vertices in S are mutually adjacent (i.e. form a complete subgraph/a clique).

I have the following algorithm (pseudo-code):

foreach vertex in S  {
  // Check that the vertex has enough edges
  if (vertex.edges.count < S.size - 1)
    return false;

  // Check that there is an edge to each vertex in S
  foreach vertex2 in S  {
    if (!vertex.hasEdgeTo(vertex2))
      return false;
  }
}
return true;

The problem is that this algorithm’s worst-case performance is O(|V|²) (in case when the subset S contains all the vertices of a complete graph).

My question is: is there a faster algorithm that runs with a better big O worst-case complexity?

Answer 1

Assuming that you can check whether two vertices are incident in O(1), your algorithm has O(|V|²) = O(|E|) time complexity in the worst case. I don’t think you can do better than O(|E|), since you should really check all the edges of the subgraph.