An Enumeration of Certain Equivalence Classes of Directed Multigraphs Having 0 to 5 Nodes with Each Node Having 2 Incoming and 2 Outgoing Arcs

有界度有向多重图的反馈弧集最小规模的上界

Upper bounds on minimum size of feedback arc set of directed multigraphs with bounded degree

本研究针对最大度不超过5的有向多重图，证明了其反馈弧集最小规模满足${\rm fas}(D)\le m/3$的紧上界，强化了Hanauer的猜想并改进了已有结果，同时给出了按顶点数计的紧界常数$c$的范围。

Packing and covering directed triangles

We prove that if a directed multigraph $D$ has at most $t$ pairwise arc-disjoint directed triangles, then there exists a set of less than $2t$ arcs in $D$ which meets all directed triangles in $D$, except in the trivial case $t=0$. This answers affirmatively a question of Tuza from 1990.

The watchman's walk problem on directed graphs

In a graph, a watchman's walk is a minimum closed dominating walk. Given a graph $G$ and a single watchman, the length of a watchman's walk in $G$ (the watchman number) is denoted by $w(G)$ and the typical goals of the watchman's walk problem is to determine $w(G)$ and find a watchman's walk in $G$. In this paper, we extend the watchman's walk problem to directed graphs. In a directed graph, we say that the watchman can only move to and see the vertices that are adjacent to him relative to outgoing arcs. That is, a watchman's walk is oriented and domination occurs in the direction of the arcs.…