[Solved] 1 Matching Graph G Subset Em C E G Edges Vertex Touches One Edges Em Recall Bipartite Grap Q37189171
1. A matching in a graph G is a subset EM C E(G) of edges such that each vertex touches at most one of the edges in EM. Recall that a bipartite graph is a graph G on two sets of vertices, V and V2, such that every edge has one endpoint in Vi and one endpoint in Vg. We sometimes write G (W, ; E) for this situation. For example: V 1 234 5 6 10 The edges in the above example consist of all the lines, whether solid or dotted; the solid lines form a matching. The bipartite maximum matching problem is to find a matching in a given bipartite graph G, which has the maximum number of edges among all matchings in G (a) (6 pts total) Prove that a maximum matching in a bipartite graph G = (VİNG; E) has size at most min{VM, IV2 (b) (6 pts total) Show how you can use an algorithm for max-flow to solve bipar- tite maximum matching on undirected simple bipartite graphs. That is, give an algorithm which, given an undirected simple bipartite graph G (V, V2; E), (1) constructs a directed, weighted graph G (which need not be bipartite) with weights w : E(G) → R as well as two vertices s, t E V(C), (2) solves max-flow for (G’, w), s, t, and (3) uses the solution for max-flow to find the maximum matching in G. Your algorithm may use any max-flow algorithm as a subroutine. (c)(7 pts total) Show the weighted graph constructed by your algorithm on the example bipartite graph above. Show transcribed image text 1. A matching in a graph G is a subset EM C E(G) of edges such that each vertex touches at most one of the edges in EM. Recall that a bipartite graph is a graph G on two sets of vertices, V and V2, such that every edge has one endpoint in Vi and one endpoint in Vg. We sometimes write G (W, ; E) for this situation. For example: V 1 234 5 6 10 The edges in the above example consist of all the lines, whether solid or dotted; the solid lines form a matching. The bipartite maximum matching problem is to find a matching in a given bipartite graph G, which has the maximum number of edges among all matchings in G
(a) (6 pts total) Prove that a maximum matching in a bipartite graph G = (VİNG; E) has size at most min{VM, IV2
(b) (6 pts total) Show how you can use an algorithm for max-flow to solve bipar- tite maximum matching on undirected simple bipartite graphs. That is, give an algorithm which, given an undirected simple bipartite graph G (V, V2; E), (1) constructs a directed, weighted graph G (which need not be bipartite) with weights w : E(G) → R as well as two vertices s, t E V(C), (2) solves max-flow for (G’, w), s, t, and (3) uses the solution for max-flow to find the maximum matching in G. Your algorithm may use any max-flow algorithm as a subroutine.
(c)(7 pts total) Show the weighted graph constructed by your algorithm on the example bipartite graph above.
Expert Answer
Answer to 1. A matching in a graph G is a subset EM C E(G) of edges such that each vertex touches at most one of the edges in EM. … . . .
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