Twin-width is a new parameter informally measuring how diverse are the neighbourhoods of the graph vertices, and it extends also to other binary relational structures, e.g. to digraphs and posets. It was introduced just very recently, in 2020 by Bonnet, Kim, Thomasse and Watrigant. One of the core results of these authors is that FO model checking on graph classes of bounded twin-width is in FPT. With that result, they also claimed that posets of bounded width have bounded twin-width, thus capturing prior result on FO model checking of posets of bounded width in FPT. However, their translation from poset width to twin-width was indirect and giving only a very loose double-exponential bound. We prove that posets of width d have twin-width at most 9d with a direct and elegant argument, and show that this bound is asymptotically tight. Specially, for posets of width 2 we prove that in the worst case their twin-width is also equal 2. These two theoretical results are complemented with straightforward algorithms to construct the respective contraction sequence for a given poset.
翻译:双曲线是一个新的参数,它非正式地测量了图形顶端的相邻区的多样性,它也延伸到了其他二元关系结构,例如写字和图示。它刚刚于2020年由Bonnet、Kim、Thomasse和Watrigant在2020年由Bonnet、Kim、Thomasse和Watrigant推出。这些作者的核心结果之一是FO模型在捆绑双曲线的图形类别上检查FPT。结果还表明,受约束宽度的外形已经捆绑了双曲线,从而获得FO模型检查FPT中被捆绑宽的外形结构的先前结果。然而,它们从表宽到双曲线的转换是间接的,仅提供了非常松散的双曲线。我们证明,宽度的模范在9d最多是双曲线上,直接和优雅的论证是双曲线。结果表明,这一界限是同样紧凑紧凑的。对于宽度2号,我们证明,在最坏的案例中,它们的双曲线结构结构结构也是相同的。