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Pré-Publication, Document De Travail Année : 2021

On betweenness in order-theoretic trees

Résumé

An order-theoretic tree is a partial order such that the set of nodes larger that any node is linearly ordered. Rooted trees, ordered by the ancestor relation, are order-theoretic trees. The ternary relation B(x, y, z) of betweenness states that y is between x and z, in some sense. For linear and partial orders, for finite and order-theoretic trees, it has been characterized in first-order or monadic second-order logic. In an order-theoretic tree, B(x, y, z) means that x < y < z or z < y < x or x < y ≤ x ⊔ z or z < y ≤ x ⊔ z where x ⊔ z is the least upper-bound of incomparable x and z. We prove that the class of the induced substructures of the betweenness structures of countable order-theoretic trees is monadic second-order definable. The proof uses partitioned probe cographs, a variant of cographs, and their known five bounds, i.e., the five minimal excluded induced subgraphs, up to isomorphism. This article links two apparently unrelated topics: cographs and order-theoretic trees.
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Dates et versions

hal-03175157 , version 1 (19-03-2021)
hal-03175157 , version 2 (24-03-2021)
hal-03175157 , version 3 (23-07-2021)
hal-03175157 , version 4 (29-11-2021)
hal-03175157 , version 5 (07-09-2022)

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  • HAL Id : hal-03175157 , version 1

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Bruno Courcelle. On betweenness in order-theoretic trees. 2021. ⟨hal-03175157v1⟩
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