Degeneracy Graphs and the Neighbourhood Problem by H.-J. Kruse

By H.-J. Kruse

A few years in the past no one may have expected that during reference to degeneracy in Linear Programming relatively a brand new box. may possibly originate. In 1976 a very easy query has been posed: within the case an severe­ aspect (EP) of a polytope is degenerate and the duty is to discover all neighbouring EP's of the degenerate EP, is it essential to confirm all uncomplicated options of the corresponding equalities process linked to the degenerate EP -in order to make sure to figure out all neighbours of this EP? this question implied one other one: Does there exists a subset of the pointed out set of uncomplicated recommendations such that it suffices to discover this sort of subset which will ascertain all neighbours? step one to unravel those questions (which are influenced within the first bankruptcy of this e-book) was once to outline a graph (called degeneracy graph) the nodes of which correspond to the fundamental recommendations. It became out that this kind of graph has a few detailed houses and with the intention to remedy the above questions to start with those homes needed to be investigated. additionally the constitution of degeneracy graphs playes hereby an immense position. as the thought of degeneracy graphs was once relatively new, it was once essential to difficult first a totally new terminology and to outline new notions. Dr.

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Non-isomorphic) axn-degeneracy graphs with U nodes. Proof (1) There exists no 2x4-degeneracy graph with 10 nodes (cf. Appendix, C3 and Tab. C3)1). 1) Appendix C4 contains a further counter-example: There exists no 2xS-degeneracy graph with 13 nodes (cf. Tab. CS). (2) There are two non-isomorphic 2x4 degeneracy graphs with 12 nodes (cf. Appendix, C3 and Tab. C3)1). Moreover, the following results from Appendix C: The nodes (or degeneracy tableaux) of 2xn-degeneracy graphs (n fixed) may be divided into classes of types such that each type of node (or type of tableau) can uniquely be assigned to a 2xn-degeneracy graph.

Proof (1) There exists no 2x4-degeneracy graph with 10 nodes (cf. Appendix, C3 and Tab. C3)1). 1) Appendix C4 contains a further counter-example: There exists no 2xS-degeneracy graph with 13 nodes (cf. Tab. CS). (2) There are two non-isomorphic 2x4 degeneracy graphs with 12 nodes (cf. Appendix, C3 and Tab. C3)1). Moreover, the following results from Appendix C: The nodes (or degeneracy tableaux) of 2xn-degeneracy graphs (n fixed) may be divided into classes of types such that each type of node (or type of tableau) can uniquely be assigned to a 2xn-degeneracy graph.

8 In the case n = (J there is only one basic form of arrangement for mini- mally laid degeneracy tableaux, since a "double diagonal form" can 1) For example both tableaux in Fig. 2 (3) are equivalent in form. However, in Fig. 2 (1) all tableaux are different in form. In Fig. 2 (2) two tableaux are equivalent in form while the form of the third is different. 43 always be obtained by an appropriate column exchange (cf. Fig. 2 (3)). Therefore,n =a fixed, all axa-degeneracy tableaux are equivalent.

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