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True or false: Suppose such a bipartite graph (i.e., with no perfect matching) has more than...

True or false: Suppose such a bipartite graph (i.e., with no perfect matching) has more than one maximum matching. Then, there must be more than one constricted set.

Provide Proof or Counter. Thanks.

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Answer #1

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ANSWER:

EXPLANATION:

Bipartite graph:

A graph G whose vertices can be divided into two disjoint sets U and V such that every edge connects a vertixe in U to one in V.

Matchig:

   A matching in a G is a set of non loop edges with no shared end points

>>The vertices incident to the edges of a matchig M are saturated by M, the others are unsaturated.

Perfect Matching:

A perfect matching is a matching that saturates every vertex.

Constricted :

If there are no perfect matchings there must be one constricted matching.

Breadth-first search (BFS): is an algorithm for traversing or searching tree or graph data structures. ... It uses the opposite strategy as depth-first search, which instead explores the highest-depth nodes first before being forced to backtrack and expand shallower nodes.

1)True because in a bipartite graph if there is no perfect matching with maximum one matching atleast must have one constrised set.

If a bipartite graph has no perfect matching, then it must contain a constricted set.

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