We introduce a new parameter, called stretch-width, that we show sits strictly between clique-width and twin-width. Unlike the reduced parameters [BKW '22], planar graphs and polynomial subdivisions do not have bounded stretch-width. This leaves open the possibility of efficient algorithms for a broad fragment of problems within Monadic Second-Order (MSO) logic on graphs of bounded stretch-width. In this direction, we prove that graphs of bounded maximum degree and bounded stretch-width have at most logarithmic treewidth. As a consequence, in classes of bounded stretch-width, Maximum Independent Set can be solved in subexponential time $2^{O(n^{4/5} \log n)}$ on $n$-vertex graphs, and, if further the maximum degree is bounded, Existential Counting Modal Logic [Pilipczuk '11] can be model-checked in polynomial time. We also give a polynomial-time $O(\text{OPT}^2)$-approximation for the stretch-width of symmetric $0,1$-matrices or ordered graphs. Somewhat unexpectedly, we prove that exponential subdivisions of bounded-degree graphs have bounded stretch-width. This allows to complement the logarithmic upper bound of treewidth with a matching lower bound. We leave as open the existence of an efficient approximation algorithm for the stretch-width of unordered graphs, if the exponential subdivisions of all graphs have bounded stretch-width, and if graphs of bounded stretch-width have logarithmic clique-width (or rank-width).

翻译：我们引入了一个新的参数，称为拉伸宽度，并证明它严格地位于团宽度与孪生宽度之间。与简化参数[BKW '22]不同，平面图和多形式细分图并不具有有界的拉伸宽度。这为在具有有界拉伸宽度的图类上，对单子二阶逻辑（MSO）中广泛问题类设计高效算法留下了可能性。在此方向上，我们证明了具有有界最大度和有界拉伸宽度的图类具有至多对数级别的树宽。由此可得，在具有有界拉伸宽度的图类中，最大独立集可在亚指数时间$2^{O(n^{4/5} \log n)}$内求解（其中$n$为顶点数）；若进一步限定最大度有界，则存在性计数模态逻辑[Pilipczuk '11]可在多项式时间内完成模型检测。我们还给出了对称$0,1$矩阵（或有序图）拉伸宽度的多项式时间$O(\text{OPT}^2)$近似算法。出乎意料的是，我们证明有界度图的指数细分图具有有界拉伸宽度。这一结果使得我们能够用匹配的下界来补充树宽的对数上界。我们留下以下公开问题：无序图的拉伸宽度是否存在高效近似算法；是否所有图的指数细分图都具有有界拉伸宽度；以及具有有界拉伸宽度的图是否具有对数的团宽度（或秩宽度）。