Let G be a bipartite graph with bipartition (X, Y). For SCX, let E, be the set of edges in G incident on some vertex in S, and let E2 be the set of edges in G incident with some vertex in N(S). Is it true in general that E1 C E2? Why? b)

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
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Solve 5b

5) A
Let M be a matching in a graph G such that G has no M-augmenting path. Show
that M is a maximum matching.
Let G be a bipartite graph with bipartition (X,Y). For S C X, let Ej be the set of
edges in G incident on some vertex in S, and let E2 be the set of edges in G incident
with some vertex in N(S). Is it true in general that E1 C E2? Why?
b)
A Prove that if G is a graph with no isolated vertices, then a'(G)+ B'(G) = n(G)
Which of the following is true ? Justify.
i)
Every tree has a perfect matching.
ii)
Every tree has at most one perfact matching.
Transcribed Image Text:5) A Let M be a matching in a graph G such that G has no M-augmenting path. Show that M is a maximum matching. Let G be a bipartite graph with bipartition (X,Y). For S C X, let Ej be the set of edges in G incident on some vertex in S, and let E2 be the set of edges in G incident with some vertex in N(S). Is it true in general that E1 C E2? Why? b) A Prove that if G is a graph with no isolated vertices, then a'(G)+ B'(G) = n(G) Which of the following is true ? Justify. i) Every tree has a perfect matching. ii) Every tree has at most one perfact matching.
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