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If V’C V, then since E’ is closed we must have E’ = y ( V’). 11). If V’ = V, then G’ is a hypomatchable non-hamiltonian spanning subgraph of G which must also be nonseparable and edge maximal with these properties, since E’ is closed and nonseparable. 12). Finally, suppose that G‘ has no perfect matching and is not hypomatchable. 10 there exist nonempty X C V’ such that G‘[V’ - X ] consists of at least 1 x1+ 1 hypomatchable components, and c’(X)- (XI is maximized, where c’(X) denotes the number of odd components of G’[V’ - XI.

We shall substitute in this case ta and ( b by any integer solution of the following linear system of inequalities: In both cases, we were able to obtain a pair of dual integer solutions x and y satisfying the complementary slackness conditions. Remark 8. In the evaluation of the running time of the proposed algorithm we consider that a sequence of operations GI,G2 and G3 is known, which constructs G starting with a vertex. The most costly procedure contained in the algorithm is the determination of a maximum weighted stable set in a bipartite graph (it is bounded by the cube of the number of vertices of the bipartite graph).

If IS1 is odd, then by induction there is 0 f Xs C S such that G[S- Xs] consists of at least IXsl + 1 hypomatchable components. Then c ( X * u X,) - (x*u xsla c ( X * )- (x*l, but G [V - ( X * U Xs)] contains fewer nonhypomatchable components than does G [ V - X * ] , a contradiction. If (SI is even, then let v be any node of S, which is not a cutnode of G [ S ]and let X ’ = X U { v } . Then C(X’)- / X ’ J= c ( X * )- Jx*J and if G [ S - { v } ] is hypomatchable, then we have contradicted the choice of X * .

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Histoire des sciences mathematiques et physiques, tomes 4-7. De Descartes a Euler (1884) by Marie M.


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