On merge-models

2026-03-27

AI Summary

This work investigates effective representations of binary relational structures with bounded merge-width that admit exact reconstruction via first-order interpretations. To this end, we introduce *merge-models*—a novel representation based on tree-ordered weakly sparse structures, constructed through merge sequences and equipped with a fixed first-order interpretation to recover the original structure. As a natural generalization of twin-models under bounded merge-width, merge-models establish an equivalence between structures of bounded twin-width and acyclic merge-models. We further show that twin-models preserve linear clique-width or clique-width up to constant factors and prove that binary structures of bounded twin-width correspond precisely to acyclic merge-models with bounded radius-$r_0$ merge-width.

Abstract

Tree-ordered weakly sparse models have recently emerged as a robust framework for representing structures in an ``almost sparse'' way, while allowing the structure to be reconstructed through a simple first-order interpretation. A prominent example is given by twin-models, which are bounded twin-width tree-ordered weakly sparse representations of structures with bounded twin-width derived from contraction sequences. In this paper, we develop this perspective further. First, we show that twin-models can be chosen such that they preserve linear clique-width or clique-width up to a constant factor. Then, we introduce \emph{merge-models}, a natural analog of twin-models for merge-width. Merge-models represent binary relational structures by tree-ordered weakly sparse structures. The original structures can then be recovered by a fixed first-order interpretation. A merge-model can be constructed from a merge sequence. Then, its radius-$r$ merge-width will be, up to a constant factor, bounded by the radius-$r$ width of the merge sequence from which it is derived. Finally, we show that twin-models arise naturally as special cases of merge-models, and that binary structures with bounded twin-width are exactly those having a loopless merge-model with bounded radius-$r_0$ merge-width (for some sufficiently large constant $r_0$).

Problem

• merge-models
• twin-models
• bounded twin-width
• merge-width
• tree-ordered weakly sparse models

Innovation

• merge-models
• twin-width
• first-order interpretation
• merge-width
• tree-ordered weakly sparse models