On merge-models

TL;DR

This paper introduces merge-models, a new framework for representing structures in a weakly sparse way, generalizing twin-models and preserving key width parameters for binary relational structures.

Contribution

It develops the concept of merge-models, showing their relation to twin-models and bounded twin-width structures, and demonstrates their ability to preserve important width parameters.

Findings

Merge-models can be constructed from merge sequences with bounded merge-width.

Twin-models are special cases of merge-models.

Bounded twin-width structures correspond to loopless merge-models with bounded radius-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 $r_0$).