Hector Buffière

Hector Buffière, Yuquan Lin, Jaroslav Nešet{ř}il et al.

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 $r_0$).

Hector Buffière, Yuquan Lin, Patrice Ossona de Mendez

We study monadic dependence of binary relational structures including at least one antisymmetric relation. Our cornerstone result gives sufficient conditions for proving that a structure is monadically dependent by only considering some of its reducts, assuming they are structurally well-behaved and compatible enough. As an application, we consider some reorientation rules preserving monadic dependence of binary structures, as well as replacement of one antisymmetric relation with bounded independence number by another. Then, we apply our main (technical) result to the study of twin-width in two ways. First, generalizing the fact that twin-width boundedness is equivalent to being expandable by a linear order into a monadically dependent class, we prove that it is also equivalent to being expandable by an oriented graph with bounded independence number (for instance by a poset with bounded width or by a tournament), and that FO-model checking is fixed-parameter tractable on such an expansion. Second, we show delineation by twin-width for some new classes of oriented graphs, including oriented split graphs and local tournaments. In all these cases, we also obtain fixed-parameter tractability of FO-model checking.