SIAM Journal on Discrete Mathematics, Volume 34, Issue 3, Page 1710-1724, January 2020.
We unify several seemingly different graph and digraph classes under one umbrella. These classes are all, broadly speaking, different generalizations of interval graphs, and include, in addition to interval graphs, adjusted interval digraphs, complements of threshold tolerance graphs (known as co-TT graphs), bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray bigraphs. (The last three classes coincide, but have been investigated in different contexts.) We show that all of the above classes are united by a common ordering characterization, the existence of a min ordering. However, because the presence or absence of reflexive relationships (loops) affects whether a graph or digraph has a min ordering, to obtain this result, we must define the graphs and digraphs to have those loops that are implied by their definitions. These have been largely ignored in previous work. We propose a common generalization of all these graph and digraph classes, namely signed-interval digraphs, characterized by the existence of a compact representation, a signed-interval model, which is a generalization of known representations of the graph classes. We show that the signed-interval digraphs are precisely those digraphs that are characterized by the existence of a min ordering when the loops implied by the model are considered part of the graph. We also offer an alternative geometric characterization of these digraphs. We show that co-TT graphs are the symmetric signed-interval digraphs, the adjusted interval digraphs are the reflexive signed-interval digraphs, and the interval graphs are the intersection of these two classes, namely, the reflexive and symmetric signed-interval digraphs. Similar results hold for bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray bigraphs.

中文翻译：

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最小有序图

SIAM离散数学杂志，第34卷，第3期，第1710-1724页，2020年1月。
我们在一个保护伞下统一了几个看似不同的图和有向图类。从广义上讲，这些类都是间隔图的不同概括，除间隔图外，还包括调整后的间隔二图，阈值公差图的补集（称为co-TT图），二分区间包含图，二分共圆弧形图和双向正交射线图。（最后三个类是重合的，但是已经在不同的上下文中进行了研究。）我们显示，所有上述类都通过一个公共的排序特征（一个最小排序的存在）组合在一起。但是，由于是否存在自反关系（循环）会影响图或有向图是否具有最小顺序，因此要获得此结果，我们必须定义图和有向图，使其具有定义所隐含的那些循环。这些在以前的工作中被很大程度上忽略了。我们提出所有这些图和有向图类的通用归纳，即有符号间隔有向图，其特征是存在紧密表示，有符号间隔模型，这是图类的已知表示的概括。我们显示出，有符号间隔有向图正是那些当模型所隐含的循环被认为是图的一部分时以最小排序的存在为特征的那些有向图。我们还提供了这些有向图的替代几何特征。我们表明，co-TT图是对称的有符号区间图，调整后的有向图是自反的有符号区间图，间隔图是这两个类的交集，即自反和对称的有符号间隔图。对于二部间隔包含图，二部同圆弧图和双向正交射线二部图，结果也相似。