Depth-First Search performance in a random digraph with geometric outdegree distribution

On Knuth's conjecture for back and forward arcs in Depth First Search in a random digraph with geometric outdegree distribution

Donald Knuth, in a draft of a coming volume of The Art of Computer Programming, has recently conjectured that in Depth-First Search of a random digraph with geometric outdegree distribution, the numbers of back and forward arcs have the same distribution. We show that this conjecture is equivalent to an equality between two generating functions defined by different recursions. Unfortunately, we have not been able so use this to prove the conjecture, which still is open, but we hope that this note will inspire others to succeed with the conjecture.

Digraph Complexity Measures and Applications in Formal Language Theory

We investigate structural complexity measures on digraphs, in particular the cycle rank. This concept is intimately related to a classical topic in formal language theory, namely the star height of regular languages. We explore this connection, and obtain several new algorithmic insights regarding both cycle rank and star height. Among other results, we show that computing the cycle rank is NP-complete, even for sparse digraphs of maximum outdegree 2. Notwithstanding, we provide both a polynomial-time approximation algorithm and an exponential-time exact algorithm for this problem. The former …

同构意义下函数有向图的多项式延迟生成

Polynomial-delay generation of functional digraphs up to isomorphism

本文提出基于反向搜索的算法，以$O(n^2)$延迟和线性空间在同构意义下生成函数有向图，先处理连通情形再推广到任意情形。

On a conjecture of Knuth about forward and back arcs

Following Janson's method, we prove a conjecture of Knuth: the numbers of forward and back arcs for the depth-first search (DFS) in a digraph with a geometric outdegree distribution have the same distribution.

Simple DFS on the Complement of a Graph and on Partially Complemented Digraphs

A complementation operation on a vertex of a digraph changes all outgoing arcs into non-arcs, and outgoing non-arcs into arcs. A partially complemented digraph $\widetilde{G}$ is a digraph obtained from a sequence of vertex complement operations on $G$. Dahlhaus et al. showed that, given an adjacency-list representation of $\widetilde{G}$, depth-first search (DFS) on $G$ can be performed in $O(n + \widetilde{m})$ time, where $n$ is the number of vertices and $\widetilde{m}$ is the number of edges in $\widetilde{G}$. To achieve this bound, their algorithm makes use of a somewhat complicated sta…

半完全有向图的循环秩的FPT算法

An FPT algorithm for cycle rank on semi-complete digraphs

针对半完全有向图和有限有向团宽的有向图，提出了参数化算法，在O(9^{(w+1)4^{w+2}}·n^2)时间内判定循环秩是否至多为w，并给出了半完全有向图最小反馈弧集问题的n^{O(w)}时间算法。