A moplex is a natural graph structure that arises when lifting Dirac's classical theorem from chordal graphs to general graphs. 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, in part, 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, when restricted to connected graphs, the class of $2$-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.

翻译：简单团（moplex）是一种自然的图结构，它源于将狄拉克经典定理从弦图推广到一般图的过程。虽然每个非完全图至少有两个简单团，但关于具有有界个数简单团的图的结构性质，目前所知甚少。研究这些图的部分动机来自于一般图中的简单团与弦图中的单纯模之间的平行关系：与简单团的情况不同，人们对具有有界个数单纯模的弦图的性质已有充分理解。例如，最多有两个单纯模的弦图是区间图。在这项工作中，我们启动了对$k$-简单团图的研究，这类图定义为包含最多$k$个简单团的图。其中特别值得关注的是最小非平凡情形$k=2$，它构成了区间图类的对应物。作为我们的主要结构结果，我们证明，当限制在连通图上时，$2$-简单团图类介于真区间图类和余可比图类之间；此外，对于遗传类而言，两个包含关系都是紧的。从复杂度理论的角度看，这自然引出一个问题：最多两个简单团的存在是否能保证足够丰富结构，从而有效解决那些在余可比图上难解但在真区间图上可解的问题？我们针对符合这一特征的两个重要问题——图同构和最大割——提出新的归约方法，否定了这一猜想。另一方面，我们证明每个连通的$2$-简单团图包含一条哈密顿路径，这推广了连通真区间图的相同性质。