CC Clique

A CC Clique, short for Connected Components Clique, is a subgraph of a larger graph constructed by representing each connected component of the original graph as a single node, and then drawing an edge between two nodes in the new graph if and only if there exists at least one edge between the corresponding connected components in the original graph. This results in a quotient graph where connectivity between the original components is preserved.

Formally, given a graph G = (V, E), we first find the connected components C1, C2, ..., Ck of G. Then, the CC Clique G' = (V', E') is defined as follows:

• V' = {C1, C2, ..., Ck} (Each vertex in G' represents a connected component of G)
• E' = {(Ci, Cj) | for some vertex u in Ci and some vertex v in Cj, (u, v) ∈ E} (An edge exists between Ci and Cj if there's at least one edge in G connecting vertices from Ci and Cj)

The CC Clique provides a simplified representation of the graph, highlighting the relationships between the connected components rather than the individual vertices. Analyzing the CC Clique can reveal higher-level structural properties of the original graph, such as the presence of bridges between components and overall connectivity patterns. The resulting graph G' is guaranteed to be a clique if and only if the original graph G consists of a single connected component.