Special ℓ1-graphs

In the first half of this chapter we follow [DePa01], where proofs of results below can be found. A graph G is an equicut graph if it admit an l 1-embedding, such that the equality holds in the left-hand side of the inequality (1.2) of chapter 1, concerning the size s(d G) of this embedding. Below s(d G) means the size of such equicut embedding. This means that, for such a graph, every S in the equality (1.1) of chapter 1 corresponds to an equicut δ(S), i.e. satisfy a S = 0 if and only if S partitions V into parts of size n 2 and n 2 , where n = |V |. Remind that a connected graph is called 2-connected (or 2-vertex-connected) if it remains connected after deletion of any vertex. Lemma 14.1 An equicut graph with at least four vertices is 2-connected.