Every 2-connected graph can be partitioned into two connected parts of given size. Every 2-connected non-bipartite graph on an even number of nodes can be partitioned into two equal parts such that the bipartite graph between them is connected. The continuity argument behind these proofs, and how generalizing this to more than two parts leads to topology. Basic notions of combinatorial topology: simplicial complexes, homeomorphism, geometric realization, baricentric subdivision.

Homotopy, homotopical equivalence, retracts, contractibility. Basic lemmas, including a characterization of contractible spaces as retracts of simplices.