Graph Theory: Lecture 10 - 3-Connected Graphs and Separation Properties, Slides of Applied Mathematics

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If G is 3 -connected and |G | > 4 then G has an edge e such that G /e is again 3 -connected.

A graph G is 3 -connected if and only if there exists a sequence G 0 , G 1 ,... , Gn of graphs such that the following properties hold. (1) G 0 = K 4 and Gn = G. (2) Gi+1 has an edge (x, y ) with d(x), d(y ) ≥ 3 and Gi = Gi+1/(x, y ), for every i < n.

If A, B ⊆ V (G ) and X ⊆ V (G ) is such that every A − B path in G contains a vertex in X , then we say the X separates the sets A and B in G.

Let G = (V , E ) be a graph and A, B ⊆ V. Then the minimum number of vertices separating A from B in G is equal to the maximum number of disjoint A − B paths in G.

Let a and b be two distinct vertices in G. Then if (a, b) 6 ∈ E , then the minimum number of vertices 6 = a, b separating a from b in G is equal to the maximum number of internally vertex disjoint a − b paths in G.

A graph is k-connected if and only if it contains k independent paths between any two vertices.