David R. Wood
Characterisations of interval graphs, comparability graphs, co-comparability graphs, permutation graphs, and split graphs in terms of linear orderings of the vertex set are presented. As an application, it is proved that interval graphs, co-comparability graphs, AT-free graphs, and split graphs have bandwidth bounded by their maximum degree.
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A unit cube in $k$ dimensional space (or \emph{$k$-cube} in short) is defined as the Cartesian product $R_1\times R_2\times...\times R_k$ where $R_i$(for $1\leq i\leq k$) is a closed interval of the form $[a_i,a_i+1]$ on the real line. A $k$-cube representation of a graph $G$ is a mapping of the vertices of $G$ to $k$-cubes such that two vertices in $G$ are adjacent if and only if their corresponding $k$-cubes have a non-empty intersection. The \emph{cubicity} of $G$, denoted as $\cubi(G)$, is the minimum $k$ such that $G$ has a $k$-cube representation. Roberts \cite{Roberts} showed that for a…
An asteroidal triple free graph is a graph such that for every independent triple of vertices no path between any two avoids the third. In a recent result from Corneil and Stacho, these graphs were characterised through a linear vertex ordering called an AT-free order. Here, we use techniques from abstract convex geometry to improve on this result by giving a vertex order characterisation with stronger structural properties and thus resolve an open question by Corneil and Stacho. These orderings are generated by a modification of BFS which runs in polynomial time. Furthermore, we give a linear…
Weak and strong coloring numbers are generalizations of the degeneracy of a graph, where for each natural number $k$, we seek a vertex ordering such every vertex can (weakly respectively strongly) reach in $k$ steps only few vertices with lower index in the ordering. Both notions capture the sparsity of a graph or a graph class, and have interesting applications in the structural and algorithmic graph theory. Recently, the first author together with McCarty and Norin observed a natural volume-based upper bound for the strong coloring numbers of intersection graphs of well-behaved objects in $\…
Consider two horizontal lines in the plane. A pair of a point on the top line and an interval on the bottom line defines a triangle between two lines. The intersection graph of such triangles is called a simple-triangle graph. This paper shows a vertex ordering characterization of simple-triangle graphs as follows: a graph is a simple-triangle graph if and only if there is a linear ordering of the vertices that contains both an alternating orientation of the graph and a transitive orientation of the complement of the graph.
Paths and Intersections: Characterization of Quasi-metrics in Directed Okamura-Seymour Instances
本文研究有向 Okamura-Seymour 图中的拟度量实现问题，证明给定边界终端循环次序时 Monge 性质为充要条件，并给出构造实例及寻找循环次序的高效算法。
We study Koebe orderings of planar graphs: vertex orderings obtained by modelling the graph as the intersection graph of pairwise internally-disjoint discs in the plane, and ordering the vertices by non-increasing radii of the associated discs. We prove that for every $d\in \mathbb{N}$, any such ordering has $d$-admissibility bounded by $O(d/\ln d)$ and weak $d$-coloring number bounded by $O(d^4 \ln d)$. This in particular shows that the $d$-admissibility of planar graphs is bounded by $O(d/\ln d)$, which asymptotically matches a known lower bound due to Dvořák and Siebertz.
We consider the graph class Grounded-L corresponding to graphs that admit an intersection representation by L-shaped curves, where additionally the topmost points of each curve are assumed to belong to a common horizontal line. We prove that Grounded-L graphs admit an equivalent characterisation in terms of vertex ordering with forbidden patterns. We also compare this class to related intersection classes, such as the grounded segment graphs, the monotone L-graphs (a.k.a. max point-tolerance graphs), or the outer-1-string graphs. We give constructions showing that these classes are all distinc…
In this paper, we define and study the new problem Simultaneous PQ-Ordering. Its input consists of a set of PQ-trees, which represent sets of circular orders of their leaves, together with a set of child-parent relations between these PQ-trees, such that the leaves of the child form a subset of the leaves of the parent. Simultaneous PQ-Ordering asks whether orders of the leaves of each of the trees can be chosen simultaneously, that is, for every child-parent relation the order chosen for the parent is an extension of the order chosen for the child. We show that Simultaneous PQ-Ordering is NP-…
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。
A unit cube in $k$ dimensions ($k$-cube) is defined as the the Cartesian product $R_1\times R_2\times...\times R_k$ where $R_i$(for $1\leq i\leq k$) is a closed interval of the form $[a_i,a_i+1]$ on the real line. A graph $G$ on $n$ nodes is said to be representable as the intersection of $k$-cubes (cube representation in $k$ dimensions) if each vertex of $G$ can be mapped to a $k$-cube such that two vertices are adjacent in $G$ if and only if their corresponding $k$-cubes have a non-empty intersection. The \emph{cubicity} of $G$ denoted as $\cubi(G)$ is the minimum $k$ for which $G$ can be re…
It was noted already in the 90s that many classic graph classes, such as interval, chordal, and bipartite graphs, can be characterized by the existence of an ordering of the vertices avoiding some ordered subgraphs, called patterns. Very recently, all the classes corresponding to patterns on three vertices (including the ones mentioned above) have been listed, and proved to be efficiently recognizable. In contrast, very little is known about patterns on four vertices. One of the few graph classes characterized by a pattern on four vertices is the class of intersection graphs of rectangles that…
Well-Quasi-Ordering Eulerian Digraphs Embeddable in Surfaces by Strong Immersion
本文证明可嵌入任意曲面的最大度至多 4 的欧拉有向图在强浸入下良拟序，并给出圆图、算法与 Erdős-Pósa 性质等应用。
Towards the Recognition of Oriented Interval Graphs
研究有向区间图识别问题，通过分析方向、团序和包含边相互影响限制的方式，给出两个约束版本识别问题的线性时间算法，改进相关二次时间算法。
In this work we investigate the complexity of some problems related to the {\em Simultaneous Embedding with Fixed Edges} (SEFE) of $k$ planar graphs and the PARTITIONED $k$-PAGE BOOK EMBEDDING (PBE-$k$) problems, which are known to be equivalent under certain conditions. While the computational complexity of SEFE for $k=2$ is still a central open question in Graph Drawing, the problem is NP-complete for $k \geq 3$ [Gassner {\em et al.}, WG '06], even if the intersection graph is the same for each pair of graphs ({\em sunflower intersection}) [Schaefer, JGAA (2013)]. We improve on these results…
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