Abstract not found

A class of finite tournaments determined by a set of "forbidden subtournaments" is wellquasi -ordered if and only if it contains no infinite antichain (a set of incomparable elements). It is not known if there is an algorithm which decides whether or not a class of finite tournaments determined by a finite set of forbidden subtournaments has this property. We prove a noneffective finiteness theorem bearing on the problem. We show that for each fixed k, there is a finite set of infinite antichains, 3 k , with the following property: if any class defined by k forbidden subtournaments contains an infinite antichain, then a cofinite subset of an element of 3 k must be such an antichain. By refining this analysis and using an earlier result giving an explicit algorithm for the case k = 1, we show that there exists an algorithm which decides whether or not a class of finite tournaments determined by two forbidden subtournaments is well-quasi-ordered. Keywords: antichain, decidability, forb...

The tournament N 5 can be obtained from the transitive tournament on . . . , 5}, with the natural order, by reversing the edges between successive vertices. Tournaments that do not have N 5 as a subtournament are said to omit N 5 . We describe the structure of tournaments that omit N 5 and use this with Kruskal's Tree Theorem to prove that this class of tournaments is well-quasi-ordered. The proof involves an encoding of the indecomposable tournaments omitting N 5 by a finite alphabet.

We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.

This paper will describe the basic architecture of the system and illustrate its flexibility with several examples. These descriptions will be accompanied by commentary on the associated design decisions, but will certainly not be exhaustive. The LINK manual fills in We would like to acknowledge the support of DIMACS and NSF grant CCR-9214487. DIMACS is a cooperative project of Rutgers University, Princeton University, AT&T Laboratories, Lucent Technologies/Bell Laboratories Innovations, and Bellcore. DIMACS is an NSF Science and Technology Center, funded under contract STC-91-19999; and also receives support from the New Jersey Commission on Science and Technology.

Tournament embedding is an order relation on the class of finite tournaments. An antichain is a set of finite tournaments that are pairwise incomparable in this ordering. We say an A # B. Those

We discuss two combinatorial problems concerning classes of finite or countable structures of combinatorial type, constrained to omit a specified finite number of constraints given by forbidden substructures. These constitute decision problems, taking the finite set of constraints as input. While these problems have been studied in a number of natural contexts, it remains far from clear whether they are decidable (perhaps in polynomial time) in their general form. Among the open problems of the final section, we point to Problem 17 as a particularly concrete one; we have given a translation of that problem into completely explicit graph theoretic terms.