Enumerating Maximal Isolated Cliques Based on Vertex-Dependent Connection Lower Bound.

In this paper, we are concerned with a problem of enumerating maximal isolated cliques (MIC) in a given undirected graph. Each target to be extracted is defined as a maximal subset X of vertices which is a clique and satisfies an isolatedness. Our isolation concept is based on the notion of variable j-coreness which imposes a connection lower bound depending on each vertex in X. Based on a standard clique enumerator, we design a depth-first algorithm for the problem. In our algorithm, we can prune many hopeless cliques from which we can never obtain our solution. It is noted that our solution MICs can be divided into two classes, fixpoint solutions and non-fixpoint solutions, where the latter ones are not trivial to detect. Based on a degree descending order of vertices, we show a theoretical property of non-fixpoint MICs and present an effective method of detecting them. Our experimental results for real world (benchmark) networks show that the proposed algorithm can work very well even for a large network with over a million vertices.