A moplex is a natural graph structure that arises when lifting Dirac's classical theorem from chordal graphs to general graphs. However, while every non-complete graph has at least two moplexes, little is known about structural properties of graphs with a bounded number of moplexes. The study of these graphs is motivated by the parallel between moplexes in general graphs and simplicial modules in chordal graphs: Unlike in the moplex setting, properties of chordal graphs with a bounded number of simplicial modules are well understood. For instance, chordal graphs having at most two simplicial modules are interval. In this work we initiate an investigation of $k$-moplex graphs, which are defined as graphs containing at most $k$ moplexes. Of particular interest is the smallest nontrivial case $k=2$, which forms a counterpart to the class of interval graphs. As our main structural result, we show that the class of connected $2$-moplex graphs is sandwiched between the classes of proper interval graphs and cocomparability graphs; moreover, both inclusions are tight for hereditary classes. From a complexity theoretic viewpoint, this leads to the natural question of whether the presence of at most two moplexes guarantees a sufficient amount of structure to efficiently solve problems that are known to be intractable on cocomparability graphs, but not on proper interval graphs. We develop new reductions that answer this question negatively for two prominent problems fitting this profile, namely Graph Isomorphism and Max-Cut. On the other hand, we prove that every connected $2$-moplex graph contains a Hamiltonian path, generalising the same property of connected proper interval graphs. Furthermore, for graphs with a higher number of moplexes, we lift the previously known result that graphs without asteroidal triples have at most two moplexes to the more general setting of larger asteroidal sets.

中文翻译：

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最多有两个 moplex 的图

Moplex 是一种自然的图结构，它是在将狄拉克经典定理从弦图提升到一般图时出现的。然而，虽然每个非完全图至少有两个 moplex，但对于具有有界 moplex 数量的图的结构特性知之甚少。对这些图的研究是由一般图中的 Moplex 和和弦图中的单纯模块之间的平行关系推动的：与在 Moplex 设置中不同，具有有限数量单纯模块的和弦图的属性是很好理解的。例如，最多具有两个单纯模的和弦图是区间图。在这项工作中，我们开始研究 $k$-moplex 图，这些图被定义为最多包含 $k$ moplex 的图。特别有趣的是最小的非平凡情况 $k=2$，它形成了区间图类的对应物。作为我们的主要结构结果，我们表明连通 $2$-moplex 图的类别夹在适当区间图和可比性图的类别之间；此外，这两个内含物对于世袭阶级来说都是紧密的。从复杂性理论的角度来看，这导致了一个自然的问题，即最多两个 moplex 的存在是否保证了足够数量的结构来有效地解决已知在可比性图上难以处理的问题，但在适当的区间图上则不然。我们开发了新的归约，对符合此配置文件的两个突出问题（即图同构和 Max-Cut）否定地回答了这个问题。另一方面，我们证明每个连接的 $2$-moplex 图都包含一个哈密顿路径，概括连接的适当区间图的相同属性。此外，对于具有更多 moplex 的图，我们将先前已知的结果（即没有小行星三元组的图最多有两个 moplex）提升到更大的小行星集的更一般设置。