The Number of Connected Components in Graphs and Its Applications

For any given graph and an integer k, the number of connected components with k vertices in the graph is investigated. For the vertex set of size n and the maximum degree , the number is bounded above by (e ) k ( 1)k . The factor k is essential, since we give the lower bound n 2 k 1 for k < 2n . We also show that all connected components with k vertices are enumerable in O n+m+ (e ) k 1 k 2 log k time with O(n+m) space, where m is the number of edges. Thus, if k is bounded above by some polynomial of n (e.g. is xed and k = O(log n)), we can check all connected components with k vertices in polynomial time of n. Under the assumption, several intractable problems (including Topological Containment (Subgraph Isomorphism), Small Minimum Degree 4 Subgraph, Bipartite Graph Embedding, Steiner Tree, and k-Minimum Spanning Tree) are put into tractable.