Question

Question 3. The graph below is called the Peterson graph. (It's very cool, for reasons that we will mention at various points during the semester.) In the Peterson graph, for each of the following, either find one or explain why it doesn't exist: (a) A minimum vertex cut that is not minimal (b) A minimum vertex cut that is minimal (c) A minimal vertex cut that isn't minimum

Answer 1

In Peterson graph, each Vertex is connected by 3 edges. Peterson graph is regular graph. Suppose if we delete one vertex from Peterson graph we get Hamiltonian graph it. Is irrelevant which vertex we delete by symmetry f once we delete one vertex we easy to find Hamiltonian cycle. if we delete one vertex from Hamiltonian graph we get Concorde graph. Actually we get a graph which has Hamiltonian path. \r\n we need to remove minimum 3 Vertex from Peterson graph. therefore Peterson graph is � connected. a) Minimum Vertex cut contouring 3 Vertex eg: for set vertex a disconnected therefore S = {e, g, b} rightarrow Vertex cut by definition of minimum vertex set is an vertex cut such that no other edge Vertex out containing fewer vertex. therefore Minimum Vertex cut is always minimal vertex cut. but minimal vertex cut is not always minimum b) S = {e, g, b} is minimum which is also minimal vertex cut.