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Vertices of the generalized transitive tournament polytype

✍ Scribed by Alberto Borobia

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 249 KB
Volume: 163
Category: Article
ISSN: 0012-365X
DOI: 10.1016/0012-365x(95)00113-b

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✦ Synopsis

A nonnegative matrix T = (t~)~= t is a generalized transitive tournament matrix (GTT matrix) ift, = 0, t~ = 1 -tjl for i ~j, and 1 ~< t~ i + t~. + tk~ ~< 2 for i,j,k pairwise distinct. An approach to the problem of characterize the set of vertices of the polytope {GTT }, of all GTT matrices of order n was the introduction by Brualdi and Hwang of the *-graph associated to each T ~ {GTT },. We introduce a new graph which generalize the *-graph. The new graph will be employed to develop a computable criterion for determine whethei any given GTT matrix of order n is or is not a vertex of {GTT},. A consequence of the criterion is that if T is a vertex of {GTT},. with small number, r, of different entries then we have strong restrictions for the possible entries of T. Namely, if r ~< 6 then the set of entries of T is equal to {0, !}, {0,], 1}, t 2 {0,~,

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