network-programminggraph-theorydiscrete-mathematicsclique

A single edge(u,v) in graph(G) could be considered as clique or not?

I'm confused in graph cliques whether an edge could be considered as a clique or not as both are adjacent to each other like in attached photo, there is no clique of 3 or more edges.

Example

enter image description here

I considered all the edges as clique as there is no clique with 3 or more vertices.

Solution

  • Yes, cliques can contain only one vertex. All nodes are by themselves a clique (a 1-clique).

    And can contain pairs of nodes connected by an edge. So all edges also describe a clique (clique made of the 2 vertices connected by this edge; so you could, by language extension, say that the edge is a clique. The pair of node surely is). All edges are a 2-clique.

    Sure, cliques become interesting only from 3-clique. 1-clique and 2-clique are boring, and just made of single nodes or single edges. But they are cliques anyway.