A clique in a simple undirected graph is a complete subgraph that is not contained in any larger complete subgraph. How many cliques are there in the graph shown below ?
A clique in a graph is a subset of vertices such that every two distinct vertices in the clique are adjacent. In simpler terms, a clique is a complete subgraph where each vertex connects directly to every other vertex in the subgraph. The task is to find all maximal cliques in the given graph. A maximal clique cannot be extended by including one more adjacent vertex, meaning it is not part of any larger clique.
Let's analyze the given graph:
From the image of the graph, we can identify the following cliques:
- Triangle \( \{a, b, g\} \): This is a clique because all vertices (a, b, g) are connected to each other.
- Triangle \( \{b, c, e\} \): This is a clique consisting of vertices b, c, and e.
- Triangle \( \{a, d, g\} \): This is a clique with vertices a, d, and g.
- Triangle \( \{d, e, f\} \): This forms a clique with vertices d, e, and f.
- Triangle \( \{c, e, f\} \): This is a clique of vertices c, e, and f.
Each triangle mentioned is a maximal clique, as adding another vertex to any of these subgraphs would break the complete connectivity required for a clique.
Therefore, the total number of cliques in the graph is 5.
This matches with option 5, confirming that this is the correct answer.
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