3.1.10. Let M and N be matchings in a graph G, with |M| > |N|. Prove that there existmatchings M' and N' in G such that |M'| = |M| – 1, |N'| = |N|+ 1, and M', N' have thesame union and intersection (as edge sets) as M, N.

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Answered: 3.1.10. Let M and N be matchings in a…

Advanced Engineering Mathematics

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Chapter2: Second-order Linear Odes

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4.1.1. Definition. A separating set or vertex cut of a graph G is a set S S V(G) such that G – S has more than one component. The connectivity of G, written «(G), is the minimum size of a vertex set S such that G – S is disconnected or has only one vertex. A graph G is k-connected if its connectivity is at least k. A graph other than a complete graph is k-connected if and only if every separating set has size at least k. We can view "k-connected" as a structural condition, while “connectivity k" is the solution of an optimization problem. what is k ( Q.) ? and why ?
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3.1.10. Let M and N be matchings in a graph G, with |M| > |N|. Prove that there exist matchings M' and N' in G such that |M'| = |M| – 1, |N'| = |N|+ 1, and M', N' have the same union and intersection (as edge sets) as M, N. %3D