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  "path": "/abs/2603.26570v1",
  "publishedAt": "2026-03-30T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Hector Buffière",
    "Yuquan Lin",
    "Jaroslav Nešet{ř}il",
    "Patrice {Ossona de Mendez}",
    "Sebastian Siebertz"
  ],
  "textContent": "**Authors:** Hector Buffière, Yuquan Lin, Jaroslav Nešet{ř}il, Patrice {Ossona de Mendez}, Sebastian Siebertz\n\nTree-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 $r_0$).",
  "title": "On merge-models"
}