The mim-width of a graph is a powerful structural parameter that, when bounded by a constant, allows several hard problems to be polynomial-time solvable - with a recent meta-theorem encompassing a large class of problems [SODA2023]. Since its introduction, several variants such as sim-width and omim-width were developed, along with a linear version of these parameters. It was recently shown that mim-width and all these variants all paraNP-hard, a consequence of the NP-hardness of distinguishing between graphs of linear mim-width at most 1211 and graphs of sim-width at least 1216 [ICALP2025]. The complexity of recognizing graphs of small width, particularly those close to $1$, remained open, despite their especially attractive algorithmic applications. In this work, we show that the width recognition problems remain NP-hard even on small widths. Specifically, after introducing the novel parameter Omim-width sandwiched between omim-width and mim-width, we show that: (1) deciding whether a graph has sim-width = 1, omim-width = 1, or Omin-width = 1 is NP-hard, and the same is true for their linear variants; (2) the problems of deciding whether mim-width $\leq$ 2 or linear mim-width $\leq$ 2 are both NP-hard. Interestingly, our reductions are relatively simple and are from the Unrooted Quartet Consistency problem, which is of great interest in computational biology but is not commonly used (if ever) in the theory of algorithms.

翻译：图的mim宽度是一种强大的结构参数，当其被常数界定时，可使若干困难问题在多项式时间内可解——近期的一项元定理涵盖了一大类问题[SODA2023]。自该参数提出以来，已发展出多种变体，如sim宽度和omim宽度，以及这些参数的线性版本。最近研究表明，mim宽度及其所有变体均为paraNP困难，这源于区分线性mim宽度至多1211的图与sim宽度至少1216的图的NP困难性[ICALP2025]。识别小宽度（尤其是接近1的宽度）图类的复杂性一直悬而未决，尽管这类图具有特别吸引人的算法应用。在本工作中，我们证明即使对于小宽度，宽度识别问题仍保持NP困难。具体而言，在引入介于omim宽度与mim宽度之间的新参数Omim宽度后，我们证明：（1）判定一个图是否具有sim宽度=1、omim宽度=1或Omin宽度=1是NP困难的，其线性变体同样如此；（2）判定mim宽度≤2或线性mim宽度≤2的问题均为NP困难。值得注意的是，我们的归约相对简洁，且源于无根四元组一致性问题，该问题在计算生物学中具有重要意义，但在算法理论中并不常用（若曾使用过）。