Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs---a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a much richer class of graphs. In particular, mixed unit interval graphs are not claw-free, compared to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyaci, Ekim, and Shalom (2017). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs. Clique-width is one of the most general structural graph parameters, where a large group of natural problems is still solvable in the tractable time when an efficient representation is given. Unfortunately, the exact computation of the clique-width representation is \NP-hard. Therefore, good upper-bounds on clique-width are highly appreciated, in particular, when such a bound is algorithmic.

中文翻译：

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混合单位区间图的 U-Bubble 模型及其应用：再谈 MaxCut 问题

区间图，实线（区间）上的段的交集图，在算法和特殊结构属性的研究中起着关键作用。单位区间图，它们的适当子类，其中每个区间都有一个单位长度，也已被广泛研究。我们研究混合单位区间图——单位区间图的推广，其中每个区间仍然有一个单位长度，但允许超过一种类型（开、闭、半闭）的区间。这个小的修改捕获了更丰富的图类。特别是，与单位区间图相比，混合单位区间图并非无爪。Heggernes、Meister 和 Papadopoulos 定义了单位区间图的表示，称为气泡模型，结果证明它在算法设计中很有用。我们将此模型扩展到混合单位区间图类，并通过提供用于解决混合单位区间图上的 MaxCut 问题的次指数时间算法来证明该广义模型的优点。此外，我们为混合单位区间图的某些子类推导出多项式时间算法。我们指出 Boyaci、Ekim 和 Shalom（2017）在证明单位区间图上 MaxCut 问题的多项式时存在重大错误。因此，这个问题在单位区间图上的时间复杂度仍然是开放的。我们进一步为混合单位区间图的集团宽度提供了更好的算法上限。Clique-width 是最通用的结构图参数之一，在给出有效表示的情况下，大量自然问题仍然可以在易处理的时间内解决。不幸的是，集团宽度表示的精确计算是\NP-hard。因此，高度赞赏集团宽度的良好上限，特别是当这样的界限是算法时。