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By Peter Zörnig

Many difficulties in economics could be formulated as linearly restricted mathematical optimization difficulties, the place the possible resolution set X represents a convex polyhedral set. In perform, the set X usually comprises degenerate verti- ces, yielding varied difficulties within the choice of an optimum resolution in addition to in postoptimal analysis.The so- referred to as degeneracy graphs characterize a great tool for des- cribing and fixing degeneracy difficulties. The research of dege- neracy graphs opens a brand new box of analysis with many theo- retical points and useful purposes. the current pu- blication pursues goals. at the one hand the speculation of degeneracy graphs is built normally, as a way to function a foundation for additional purposes. however dege- neracy graphs can be used to give an explanation for simplex biking, i.e. useful and enough stipulations for biking might be de- rived.

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Thus every degeneracy graph contains at least one cycle (cf. 9). 6: Let a a x n-degeneracy graph Gy be givens1 . Any two different nodes I, I' of Gy can be connected by two disjoint paths. Proof: Without loss of generality we may assume I = {i1,"" iC7 } with 1 ~ i1 < ...

1. For the diameter d of Gy it holds that d = 2 < min{3,3} = 3. 2 represents an important result in the theory of degeneracy graphs. It means that between any two nodes of a degeneracy graph a path of length::; min{ u, n} exists. 2 is the assertion that (general) u x n-degeneracy graphs are always connected (cf. 78 The following theorem provides a formula for the number of nodes of arbitrary degeneracy graphs. tems are biuniquely assigned to degeneracy graphs (d. Def. 6). 4: Let a u x n-degeneracy graph (8) and its representation system T = 1)(8) = {t 1 , • •• , t q } be given.

1: Let a u x n-degeneracy graph Gy be given (Y E JRuXn)74. Between any two nodes I, l' of Gy there exists a path75 of length P = 11\1'1. The following proof is by induction (cf. Beisel/Mendel (1987:8)): We assume that the assertion holds for P = po with po 2:: 1. (The case p = 1 is trivial, since 1,1' are connected by an edge). Without loss of generality we may further assume I = {iI, ... ,ju}, I' = {jp+l, ... ,ju,ju+I, ... ,ju+p}, where jl, ... ,ju+p E {l, ... ,N} denote different indices (N = n + u).

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