By Reinhard Diestel

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Let U be its set of degree 3 vertices. Let C be the set of all cycles in G that avoid U and meet H in exactly one vertex. Let Z ⊆ V (H) U be the set of those vertices. For each z ∈ Z pick a cycle Cz ∈ C that meets H in z, and put C := { Cz | z ∈ Z }. By the maximality of H, the cycles in C are disjoint. Let D be the set of the 2-regular components of H that avoid Z. Then C ∪ D is another set of disjoint cycles. If |C ∪ D| k, we are done. Otherwise we can add to Z one vertex from each cycle in D to obtain a set X of at most k − 1 vertices that meets all the cycles in C and all the 2-regular components of H.

Since M is a maximal matching, it contains an edge a b with a = a or b = b . In fact, we may assume that a = a : for if a is unmatched (and b = b ), then ab is an alternating path, and so the end of a b ∈ M chosen for U was the vertex b = b. Now if a = a is not in U , then b ∈ U , and some alternating path P ends in b . But then there is also an alternating path P ending in b: either P := P b (if b ∈ P ) or P := P b a b. By the maximality of M , however, P is not an augmenting path. So b must be matched, and was chosen for U from the edge of M containing it.

The Basics If G = M X is a subgraph of another graph Y , we call X a minor of Y and write X Y . Note that every subgraph of a graph is also its minor; in particular, every graph is its own minor. 1, any minor of a graph can be obtained from it by ﬁrst deleting some vertices and edges, and then contracting some further edges. 3). 8 If G = T X is the subgraph of another graph Y , then X is a topological topological minor of Y (Fig. 3). minor minor; G X Y Fig. 3. 3 ] If G = T X, we view V (X) as a subset of V (G) and call these vertices the branch vertices of G; the other vertices of G are its subdividing vertices.