[Solved] Let G Directed Graph Vertex Decomposition Graphs Collection U1 U2 Sets V S Ui Ui Uj 6 J Q37179576

Let G be a directed graph. A vertex decomposition of a graphs isa collection U1 ∪ U2 . . . of sets, so that V = S Ui and Ui ∩ Uj =∅ for i 6= j. A Hamilton decomposition is a decomposition so thatG(Ui) has an Hamiltonian path for every i.

1. Show that such a decomposition exists if an only if we canselect a subset of the edges so that the indegree of every vertexand the outdegree of every vertex is 1. (You have to prove twothings here).

2. Give an algorithm that checks if such a decomposition exists.(Hint: You have to double the vertices to get a bipartite graph (V,V, E0 ) with E0 like E. Choose the appropriate capacities.

Please write time complexity of the algorithm.

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Answer to Let G be a directed graph. A vertex decomposition of a graphs is a collection U1 ∪ U2 . . . of sets, so that V = S Ui … . . .