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A Decision Problem Involving Tournaments
, 1996
"... 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 ..."
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Cited by 5 (2 self)
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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 wellquasiordered. Keywords: antichain, decidability, forb...
Tournaments that omit N 5 are wellquasiordered
, 2002
"... 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 th ..."
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Cited by 2 (1 self)
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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 wellquasiordered. The proof involves an encoding of the indecomposable tournaments omitting N 5 by a finite alphabet.
Computers and Discovery in Algebraic Graph Theory
 Edinburgh, 2001), Linear Algebra Appl
, 2001
"... We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory. ..."
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Cited by 1 (0 self)
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We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.
LINK: A system for graph computation
, 1999
"... 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 ..."
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Cited by 1 (0 self)
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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 CCR9214487. 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 STC9119999; and also receives support from the New Jersey Commission on Science and Technology.
A Classification of Antichains of Finite Tournaments
, 2002
"... 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 ..."
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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
Forbidden Substructures and Combinatorial Dichotomies: WQO and Universality
, 2009
"... 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 th ..."
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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.