A non-empty set $X$ of vertices of an acyclic digraph is called connected if the underlying undirected graph induced by $X$ is connected and it is called convex if no two vertices of $X$ are connected by a directed path in which some vertices are not in $X$. The set of convex sets (connected convex sets) of an acyclic digraph $D$ is denoted by $\sco(D)$ ($\scc(D)$) and its size by $\co(D)$ ($\cc(D)$). Gutin, Johnstone, Reddington, Scott, Soleimanfallah, and Yeo (Proc. ACiD'07) conjectured that the sum of the sizes of all (connected) convex sets in $D$ equals $Θ(n \cdot \co(D))$ ($Θ(n \cdot \cc(D))$) where $n$ is the order of $D$.
In this paper we exhibit a family of connected acyclic digraphs with $\sum_{C\in \sco(D)}|C| = o(n\cdot \co(D))$ and $\sum_{C\in \scc(D)}|C| = o(n\cdot \cc(D))$. We also show that the number of connected convex sets of order $k$ in any connected acyclic digraph of order $n$ is at least $n-k+1$. This is a strengthening of a theorem by Gutin and Yeo.
Perfect phylogenies via branchings in acyclic digraphs and a generalization of Dilworth's theorem
受癌症基因组学应用的启发，并遵循Hajirasouliha和Raphael的工作（WABI 2014），Hujdurović et al.（IEEE TCBB，即将出现）介绍了最小无冲突行分割（MCRS）问题：将给定二进制矩阵的每一行拆分为一组行的按位或，使得所得矩阵对应于完美的同构，并且在所有矩阵中具有最小可能的行数与此属性。Hajirasouliha和Raphael也提出了类似问题的研究，其中的任务是最小化所得矩阵的不同行的数量。Hujdurović等人证明了这两个问题都是NP-难的，给出了传递可定向图的相关刻画，并提出了一个基于着色协可比图的MCRS问题的多项式时间启发式算法。我们给出了新的，更透明的配方的两个问题，表明这些问题是等价于两个优化问题的分支在派生的有向无环图。在这些公式的基础上，我们得到了关于这两个问题的新结果，包括：（i）Hujdurović等人通过推广Dilworth定理的有向图中的新的最小-最大结果来加强启发式，这可能是独立的兴趣，（ii）两个问题的APX-硬度结果，（iii）近似算法，和（iv）指数时间算法解决这两个问题的最优性比天真的蛮力方法更快。我们的工作涉及到几个很好的研究概念在组合优化：链分区在偏序集，层状超图，（经典和加权）着色的图形。
有向图$D$的顶点的非空子集$S$称为{ \it安全集}，如果\begin{itemize} \item[(i)] for every strongly connected component $M$ of $D-S$, there exists a strongly connected component $N$ of $D[S]$ such that there exists an arc from $M$ to $N$; and \item[(ii)] for every strongly connected component $M$ of $D-S$ and every strongly connected component $N$ of $D[S]$, we have $|M|\leq |N|$ whenever there exists an arc from $M$ to $N$. \end{itemize}在无圈有向图的情况下，顶点的集合$X$是安全集，精确地当$X$是{ \it内支配集}时，也就是说，不在$X$中的每个顶点至少有一个到$X$的弧。我们证明了，即使是可追踪的无圈有向图（有一条哈密尔顿路径），也是NP-难找到的-支配集的最小基数。然后，我们证明了这个问题也是NP-难的比赛，并给出了一个多项式算法，为每一个正常数$c$，找到一个最小的基数安全集在$n…
Algorithms and hardness for Metric Dimension on digraphs
Université Clermont Auvergne, CNRS, Clermont Auvergne INP, Mines Saint-Étienne, LIMOS（克莱蒙奥弗涅大学，法国国家科学研究中心，克莱蒙奥弗涅理工学院，圣艾蒂安国立高等矿业学校，LIMOS）；Department of Mathematics and Statistics, University of Turku（图尔库大学数学与统计系）；Turku Collegium for Science, Medicine and Technology TCSMT（图尔库科学技术医学学院）
