This chapter discusses an idea that might suggest some non-trivial approaches to NP-complete problems. The problem of computing the independence number of a graph is NP-complete. The matching number, on the other hand is computable in polynomial time. This difference in their computational complexity implies that, to attack these two problems, different strategies have to be applied. The chapter surveys some methods to obtain upper bounds on the independence number α(G) of a graph. The chapter points out that complexity consideration concerning the independence number problem motivate, and may even initiate, research in fields like algebraic geometry, linear algebra and algebraic topology.
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