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Well quasi-order and atomicity for combinatorial structures under consecutive orders

Victoria Ironmonger, Nik Ruškuc

TL;DR

This paper develops a general framework to decide well quasi-ordering and atomicity for posets of combinatorial structures under consecutive orders, unifying graphs, digraphs, relations, and, via word-encodings, even certain permutations. Central to the approach are the notions of valid and bountiful structures and the associated factor graphs, which translate structure containment into path-embedding questions; for bountiful types, the $wqo$ problem reduces to finiteness of the avoidance set, while atomicity reduces to joint-embedding-type criteria in the factor graph (often captured by strong connectivity or bicycle structure). The authors obtain decidability results for broad classes (including $\, ext{G}, \, ext{S}, \, ext{D}, \, ext{T}, \, ext{R}_{\sigma}$) and for words, with a specialized treatment of the invalid double-ascents by linking them to word-avoiding problems. Collectively, these results provide algorithmic decision procedures for $wqo$ and atomicity across many familiar combinatorial families, and outline a path toward handling intermediate and invalid structure types within the same digraph-encoding paradigm.

Abstract

We consider partially ordered sets of combinatorial structures under consecutive orders, meaning that two structures are related when one embeds in the other such that `consecutive' elements remain consecutive in the image. Given such a partially ordered set, we may ask decidability questions about its avoidance sets: subsets defined by a finite number of forbidden substructures. Two such questions ask, given a finite set of structures, whether its avoidance set is well quasi-ordered (i.e. contains no infinite antichains) or atomic (i.e. cannot be expressed as the union of two proper subsets). Extending some recent new approaches, we will establish a general framework, which enables us to answer these problems for a wide class of combinatorial structures, including graphs, digraphs and collections of relations.

Well quasi-order and atomicity for combinatorial structures under consecutive orders

TL;DR

This paper develops a general framework to decide well quasi-ordering and atomicity for posets of combinatorial structures under consecutive orders, unifying graphs, digraphs, relations, and, via word-encodings, even certain permutations. Central to the approach are the notions of valid and bountiful structures and the associated factor graphs, which translate structure containment into path-embedding questions; for bountiful types, the problem reduces to finiteness of the avoidance set, while atomicity reduces to joint-embedding-type criteria in the factor graph (often captured by strong connectivity or bicycle structure). The authors obtain decidability results for broad classes (including ) and for words, with a specialized treatment of the invalid double-ascents by linking them to word-avoiding problems. Collectively, these results provide algorithmic decision procedures for and atomicity across many familiar combinatorial families, and outline a path toward handling intermediate and invalid structure types within the same digraph-encoding paradigm.

Abstract

We consider partially ordered sets of combinatorial structures under consecutive orders, meaning that two structures are related when one embeds in the other such that `consecutive' elements remain consecutive in the image. Given such a partially ordered set, we may ask decidability questions about its avoidance sets: subsets defined by a finite number of forbidden substructures. Two such questions ask, given a finite set of structures, whether its avoidance set is well quasi-ordered (i.e. contains no infinite antichains) or atomic (i.e. cannot be expressed as the union of two proper subsets). Extending some recent new approaches, we will establish a general framework, which enables us to answer these problems for a wide class of combinatorial structures, including graphs, digraphs and collections of relations.

Paper Structure

This paper contains 14 sections, 45 theorems, 2 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and results from graph theory
  4. Factor Graphs
  5. Valid structures
  6. Bountiful structures
  7. Well quasi-order for bountiful structures
  8. Atomicity for Bountiful Structures
  9. Words under the consecutive order
  10. An invalid example: double ascents
  11. Double ascents and words
  12. WQO for double ascents
  13. Atomicity for double ascents
  14. Concluding remarks and open problems

Key Result

Proposition 2.9

A subset $Y$ of a poset $(X, \leq)$ is atomic if and only if for any $x, y \in Y$ there exists $z \in Y$ such that $x, y \leq z$; this property is know as the joint embedding property (JEP). When $x, y \leq z$, we will say that $z$joins$x$ and $y$, or simply that $x$ and $y$join.

Figures (8)

  • Figure 1: The graphs $G$, $H$ and $G\space\! \upharpoonright_{[2, 3]}$ respectively from Example \ref{['ex intro']}.
  • Figure 2: Various examples of bicycles.
  • Figure 3: The factor graph from Example \ref{["ex factor graphs don't work"]} whose paths can create cycles.
  • Figure 4: The graph corresponding to the path $\pi$ in Example \ref{["ex factor graphs don't work"]}
  • Figure 5: $\Gamma_K$ from Examples \ref{['ex d ascent non wqo']} and \ref{['ex da atomicity 2']}.
  • ...and 3 more figures

Theorems & Definitions (126)

  • Example 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Definition 2.7
  • Definition 2.8
  • Proposition 2.9
  • Example 2.10
  • ...and 116 more