区间图的部分表示的扩展

单位圆弧图的部分表示的扩展

Extending Partial Representations of Unit Circular-arc Graphs

Klavík等人（2011）提出的部分表示扩展问题（partial representation extension problem）将识别问题推广。在这个简短的说明中，我们证明了这个问题是NP-完全的单位圆弧图。

Extending partial automorphisms of $n$-partite tournaments

We prove that for every $n\geq 2$ the class of all finite $n$-partite tournaments (orientations of complete $n$-partite graphs) has the extension property for partial automorphisms, that is, for every finite $n$-partite tournament $G$ there is a finite $n$-partite tournament $H$ such that every isomorphism of induced subgraphs of $G$ extends to an automorphism of $H$. Our constructions are purely combinatorial (whereas many earlier EPPA results use deep results from group theory) and extend to other classes such as the class of all finite semi-generic tournaments.

Extending Partial Representations of Circular-Arc Graphs

The partial representation extension problem generalizes the recognition problem for classes of graphs defined in terms of vertex representations. We exhibit circular-arc graphs as the first example of a graph class where the recognition is polynomially solvable while the representation extension problem is NP-complete. In this setting, several arcs are predrawn and we ask whether this partial representation can be completed. We complement this hardness argument with tractability results of the representation extension problem on various subclasses of circular-arc graphs, most notably on all v…

Ramsey properties and extending partial automorphisms for classes of finite structures

We show that every free amalgamation class of finite structures with relations and (symmetric) partial functions is a Ramsey class when enriched by a free linear ordering of vertices. This is a common strengthening of the Nešetřil-Rödl Theorem and the second and third authors' Ramsey theorem for finite models (that is, structures with both relations and functions). We also find subclasses with the ordering property. For languages with relational symbols and unary functions we also show the extension property for partial automorphisms (EPPA) of free amalgamation classes. These general results s…

区间嵌套有向图的特征

A characterization of interval nest digraphs

本文通过嵌套序列为特征，完整刻画了区间嵌套有向图的结构特性。

Extending Partial Orthogonal Drawings

We study the planar orthogonal drawing style within the framework of partial representation extension. Let $(G,H,Γ_H )$ be a partial orthogonal drawing, i.e., G is a graph, $H\subseteq G$ is a subgraph and $Γ_H$ is a planar orthogonal drawing of H. We show that the existence of an orthogonal drawing $Γ_G$ of $G$ that extends $Γ_H$ can be tested in linear time. If such a drawing exists, then there also is one that uses $O(|V(H)|)$ bends per edge. On the other hand, we show that it is NP-complete to find an extension that minimizes the number of bends or has a fixed number of bends per edge.

判定Robinson矩阵的一个结构特征

A Structural Characterization for Certifying Robinsonian Matrices

一个对称矩阵是罗宾逊的，如果它的行和列可以同时重新排序，使得当向对角线移动时，元素在行和列中单调非减。当一个图是单位区间图时，它的邻接矩阵恰好是Robinson的，从而Robinson矩阵构成了单位区间图类的矩阵模拟。在这里，我们提供了一个结构表征的Robinson矩阵的禁止子结构，扩展的概念，加权图的星形三元组。这意味着已知的单位区间图的特征，并导致一个有效的算法来证明矩阵是不是罗宾逊。

线图序区间的普适性

Universality of intervals of line graph order

证明了对每一个$d\geq 3$，具有最大度的有限图的线图类$d$的同态阶是泛的.这意味着每个有限或可数无限偏序集都可以用最大度为$d$的图的线图来表示，这些图是通过同态的存在来排序的。

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

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。

Extending Partial Representations of Rectangular Duals with Given Contact Orientations

A rectangular dual of a graph $G$ is a contact representation of $G$ by axis-aligned rectangles such that (i)~no four rectangles share a point and (ii)~the union of all rectangles is a rectangle. The partial representation extension problem for rectangular duals asks whether a given partial rectangular dual can be extended to a rectangular dual, that is, whether there exists a rectangular dual where some vertices are represented by prescribed rectangles. Combinatorially, a rectangular dual can be described by a regular edge labeling (REL), which determines the orientations of the rectangle con…

关于有限偏序的可堆性

On the heapability of finite partial orders

本文研究将有限偏序集划分为最少可堆子集的问题，给出类似Dilworth定理的刻画及流算法，并对区间情形提出贪心算法，最后提出相关开放问题。

梯形图的图同构完全性

Graph isomorphism completeness for trapezoid graphs

梯形图的图同构问题的复杂性已经公开了十多年。本文证明了该问题是GI-完全的。更确切地说，我们证明了图同构问题是GI-完全的可比图的偏序集的区间维数为2和高度为3。与此相反，已知的问题是可解决的多项式时间的可比性图的偏序集的区间维数最多为2和高度最多为2。

用于避免离群点的部分k-means、数学规划形式化与复杂性结果

Partial k-means to avoid outliers, mathematical programming formulations, complexity results

本文提出部分k-means变体PMSSC，通过移除固定数量离群点来缓解k-means对离群点敏感的问题，给出整数规划形式化与复杂性结果，并在一维情形下用动态规划算法求解无权PMSSC。