简单三角图与线性区间阶的识别是多项式的
The Recognition of Simple-Triangle Graphs and of Linear-Interval Orders is Polynomial
摘要：
几何对象的交图由于其有趣的结构和众多的应用而被广泛研究;突出的例子包括区间图和置换图。本文研究了一类推广了区间图和置换图的自然图，即\emph{simple-triangle}图。简单三角形图-也称为\emph{PI}图（点-间隔）-是由线$L_{1}$上的点和平行线$L_{2}$上的间隔定义的三角形的相交图。它们自然地位于排列图和梯形图之间，排列图和梯形图分别是$L_{1}$和$L_{2}$之间的线段的交图和$L_{1}$和$L_{2}$之间的多面体的交图。虽然各种有效的识别算法的排列和梯形图是众所周知的存在，简单三角形图的识别一直是一个开放的问题，因为他们介绍了Corneil和Kamula三十年前。在本文中，我们解决了这个问题，证明了简单三角形图可以在多项式时间内识别。因此，我们的算法还解决了偏序领域长期存在的开放问题，即\emph{linear-interval orders}的识别，即偏序$P=P_{1}\cap P_{2}$的识别，其中$P_{1}$是线性序，$P_{2}$是区间序。这是关于识别偏序$P$的第一个结果，偏序$P$是来自两个不同类$\mathcal{P}_{1}$和$\mathcal{P}_{2}$的序的交集。与此完全相反，偏序$P$是交叉的订单从同一类$\mathcal{P}$已被广泛研究，在大多数情况下，这些识别问题的复杂性状态已经建立。
英文摘要：
Intersection graphs of geometric objects have been extensively studied, both due to their interesting structure and their numerous applications; prominent examples include interval graphs and permutation graphs. In this paper we study a natural graph class that generalizes both interval and permutation graphs, namely \emph{simple-triangle} graphs. Simple-triangle graphs - also known as \emph{PI} graphs (for Point-Interval) - are the intersection graphs of triangles that are defined by a point on a line $L_{1}$ and an interval on a parallel line $L_{2}$. They lie naturally between permutation and trapezoid graphs, which are the intersection graphs of line segments between $L_{1}$ and $L_{2}$ and of trapezoids between $L_{1}$ and $L_{2}$, respectively. Although various efficient recognition algorithms for permutation and trapezoid graphs are well known to exist, the recognition of simple-triangle graphs has remained an open problem since their introduction by Corneil and Kamula three decades ago. In this paper we resolve this problem by proving that simple-triangle graphs can be recognized in polynomial time. As a consequence, our algorithm also solves a longstanding open problem in the area of partial orders, namely the recognition of \emph{linear-interval orders}, i.e. of partial orders $P=P_{1}\cap P_{2}$, where $P_{1}$ is a linear order and $P_{2}$ is an interval order. This is one of the first results on recognizing partial orders $P$ that are the intersection of orders from two different classes $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$. In complete contrast to this, partial orders $P$ which are the intersection of orders from the same class $\mathcal{P}$ have been extensively investigated, and in most cases the complexity status of these recognition problems has been already established.
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