 
Summary: Short certificates for tournaments
Noga Alon
Mikl´os Ruszink´o
February 22, 2002
Abstract
An isomorphism certificate of a labeled tournament T is a labeled subdigraph of T which to
gether with an unlabeled copy of T allows the errorless reconstruction of T. It is shown that any
tournament on n vertices contains an isomorphism certificate with at most n log2 n edges. This
answers a question of Fishburn, Kim and Tetali. A score certificate of T is a labeled subdigraph of
T which together with the score sequence of T allows its errorless reconstruction. It is shown that
there is an absolute constant > 0 so that any tournament on n vertices contains a score certificate
with at most (1/2  )n2
edges.
1 Introduction
A tournament is an oriented complete graph. An isomorphism certificate of a labeled tournament
T is a labeled subdigraph D of T which together with an unlabeled copy of T allows the errorless
reconstruction of T. More precisely, if V = {v1, . . . , vn} denotes the vertex set of T, then a subdigraph
D of T is such a certificate if for any tournament T on V which is isomorphic to T and contains D, T
is, in fact, identical to T. The size of the certificate D is the number of its edges, and D is a minimum
certificate if no isomorphism certificate has a smaller size.
