By Peter Zörnig

Many difficulties in economics could be formulated as linearly restricted mathematical optimization difficulties, the place the possible answer set X represents a convex polyhedral set. In perform, the set X usually includes degenerate verti- ces, yielding assorted difficulties within the choice of an optimum answer in addition to in postoptimal analysis.The so- known as degeneracy graphs characterize a great tool for des- cribing and fixing degeneracy difficulties. The learn of dege- neracy graphs opens a brand new box of study with many theo- retical facets and useful purposes. the current pu- blication pursues goals. at the one hand the speculation of degeneracy graphs is built more often than not, that allows you to function a foundation for additional functions. nonetheless dege- neracy graphs could be used to give an explanation for simplex biking, i.e. useful and adequate stipulations for biking should be de- rived.

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**Additional info for Degeneracy Graphs and Simplex Cycling**

**Example text**

We obtain (cf. Fig. 2): N = 8, u = 4, q = 3, 80 cr. g. Aigner (1975:249). 48 Nl = 7, N2 = 5, N3 = 5, N 1 ,2 = 4, N 1 ,3 = 4, N 2,3 = 3, N 1 ,2,3 = 2. ) = 70 - 35 - 5 - 5 + 1 + 1 + 0 - 0 = 27. Fig. 4 permits efficient computation of the node number U of degeneracy graphs. ) = 70 4 x 4-submatrices of (YII4 ) would have to be checked with regard to regularity. The following statements refer to the connectivity of (general) degeneracy graphs. We will show that any two nodes of a degeneracy graph can be connected by at least two disjoint paths.

We are now able to state an upper bound for the diameter of degeneracy graphs. 2: Let Gy be a u x n-degeneracy graph. For the diameter d = d( Gy) of Gy holds that d ~ min{u,n}. 73 Cf. Appendix, Def. 6. 74 Cf. 6. 75 Cf. Appendix, Def. BA. 76 Cf. g. Kowalsky (1975:37). 77 Cf. 1. 45 I, I' denote any two nodes of G y and p = II\ll. From III = III = u it follows that p ::; u. On the other hand p + u ::; n + u, Proof: Let thus p::; n and finally p::; min{u,n}. • Obviously the diameter of the 4 x 2-degeneracy graph in Fig.

Basic indices column indices nonbasic columns basic solution The example above demonstrates that for degenerate vertices "information will be lost" to some extent, if a polytope is presented by the graph of a polytope (cf. Def. 2). On the other hand the representation graph reflects the structure of all vertices, even the degenerate ones. 4) be given again. Moreover, let xO E X be a a-degenerate vertex with the corresponding b~sis set BO. The following graphs represent the structure of xO (cf.