Cliques and Independent Sets

1. Cliques and Independent Sets prepared and Instructed by Shmuel Wimer Eng. Faculty, Bar-Ilan University Cliques and Independent Sets

2. Shannon Capacity A message consisting of signals belonging to a certain finite alphabet A is transmitted over a noisy channel. The message is a sequence of words of k signals each. Some pairs of signals are so similar that they can be confounded by the receiver due to noise. What is the largest number of distinct words that can be used in messages without a confusion at the receiver? Example. A={0,1,2,3,4}. k=2. If the noise results errors of i+1 and i-1 (mod 5), the message 00, 12, 24, 31, 43, can safely be transmitted. Cliques and Independent Sets

3. Definition. The strong product of two graphs G and H is defined by the vertex set V(G) x V(H). Two vertices ux and vy are adjacent iff and x=y, or and u=v,or and . 0 0 0 4 4 4 1 1 3 3 3 2 2 2 The strong product is embedded on a torus. 1 0 0 1 2 3 4 0 Cliques and Independent Sets

4. Let G be the graph with vertex set A. An edge uv is defined if u and v represent signals that might be confused with each other. Gk is a strong product of k copies of G, representing words of length k over A. Q: What are the edges of Gk ? A: Two distinct words (u1,u2,…,uk) and (v1,v2,…,vk) are connected with an edge if either ui=vi or for 1≤i≤k. Edges of Gk correspond to words that might be confused with each other. The largest number of distinct words equals the size of maximum independent set α(Gk). Cliques and Independent Sets

5. In this example k=2 and α(C52)=5. 0 4 3 2 1 0 0 1 2 3 4 0 Cliques and Independent Sets

6. Digraphs and Kernels A stable vertexsetS in a digraph D is a stable set in its underlying graph G. If S is maximal then every vertex of G - S is adjacent to S. For digraph it is natural to replace the adjacency by dominance. A kernel in . Cliques and Independent Sets