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 $r_0$).

翻译：树序弱稀疏模型近期作为一种鲁棒框架出现，能够以“几乎稀疏”的方式表示结构，同时允许通过简单的一阶解释重建结构。一个典型例子是孪生模型——它是从收缩序列导出的、具有有界孪生宽的树序弱稀疏结构表示。本文进一步深化这一视角。首先，我们证明孪生模型可以选择为保持线性团宽或团宽至多常数倍的形式。其次，我们引入合并模型（merge-models），即针对合并宽度的孪生模型自然类比。合并模型通过树序弱稀疏结构表示二元关系结构，原始结构可通过固定的一阶解释恢复。合并模型可由合并序列构造，其半径r合并宽度至多常数倍地受限于导出它的合并序列的半径r宽度。最后，我们证明孪生模型自然地作为合并模型的特例出现，且具有有界孪生宽的二元结构恰好是那些具有无环合并模型且半径r0合并宽度有界（对于足够大的常数r0）的结构。