Heroes in orientations of chordal graphs

弦图是完全可定向的

Chordal Graphs are Fully Orientable

设D是图G的无圈定向。如果D的弧的反转产生一个有向环，则称之为从属弧。设m和M表示G的所有非循环定向上的依赖弧数的最小值和最大值。我们称G是完全可定向的，如果G有一个非循环定向，其中对任意d满足m <= d <= M，恰有d条依赖弧.一个图G称为弦图，如果G中每个长度至少为4的圈都有一个弦。我们证明了所有弦图都是完全可定向的。

关于禁止诱导森林的有向图中的英雄

On heroes in digraphs with forbidden induced forests

本文研究禁止诱导森林的有向图类的英雄，刻画了不同度定向星及禁止定向扫帚情形下能保证二色数有界的英雄结构。

弦图的二色数

Dichromatic number of chordal graphs

本文研究弦图超定向的二色数，给出其上下界，并展示一类余图定向的二色数等于基础图团数。

Combinatorial generation via permutation languages. V. Acyclic orientations

In 1993, Savage, Squire, and West described an inductive construction for generating every acyclic orientation of a chordal graph exactly once, flipping one arc at a time. We provide two generalizations of this result. Firstly, we describe Gray codes for acyclic orientations of hypergraphs that satisfy a simple ordering condition, which generalizes the notion of perfect elimination order of graphs. This unifies the Savage-Squire-West construction with a recent algorithm for generating elimination trees of chordal graphs. Secondly, we consider quotients of lattices of acyclic orientations of ch…

On the proper orientation number of chordal graphs

An orientation $D$ of a graph $G=(V,E)$ is a digraph obtained from $G$ by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each $v \in V(G)$, the indegree of $v$ in $D$, denoted by $d^-_D(v)$, is the number of arcs with head $v$ in $D$. An orientation $D$ of $G$ is proper if $d^-_D(u)\neq d^-_D(v)$, for all $uv\in E(G)$. An orientation with maximum indegree at most $k$ is called a $k$-orientation. The proper orientation number of $G$, denoted by $\overrightarrowχ(G)$, is the minimum integer $k$ such that $G$ admits a proper $k$-orientation. We prove t…

排列图、区间图和良划分弦图中的最大边开放打包

Maximum Edge Open Packing in Permutation, Interval, and Well-Partitioned Chordal Graphs

本文针对排列图、区间图和良划分弦图的最大边开放打包问题，通过定向星冲突图的最大团或划分树动态规划，分别给出\textit{O}(n^4)时间的算法。