Tourist中的反馈顶点集

Counting orientations of graphs with no strongly connected tournaments

Let $S_k(n)$ be the maximum number of orientations of an $n$-vertex graph $G$ in which no copy of $K_k$ is strongly connected. For all integers $n$, $k\geq 4$ where $n\geq 5$ or $k\geq 5$, we prove that $S_k(n) = 2^{t_{k-1}(n)}$, where $t_{k-1}(n)$ is the number of edges of the $n$-vertex $(k-1)$-partite Turán graph $T_{k-1}(n)$, and that $T_{k-1}(n)$ is the only $n$-vertex graph with this number of orientations. Furthermore, $S_4(4) = 40$ and this maximality is achieved only by $K_4$.

7-泛锦标赛的新上界

A New Upper Bound for 7-Universal Tournaments

本文构造了一个14顶点的显式锦标赛，包含所有7顶点锦标赛作为诱导子锦标赛，通过穷举验证，将上界从15改进至14，从而确定最小阶数为13或14。

Locating Dominating Sets in local tournaments

A dominating set in a directed graph is a set of vertices $S$ such that all the vertices that do not belong to $S$ have an in-neighbour in $S$. A locating set $S$ is a set of vertices such that all the vertices that do not belong to $S$ are characterized uniquely by the in-neighbours they have in $S$, i.e. for every two vertices $u$ and $v$ that are not in $S$, there exists a vertex $s\in S$ that dominates exactly one of them. The size of a smallest set of a directed graph $D$ which is both locating and dominating is denoted by $γ^{LD}(D)$. Foucaud, Heydarshahi and Parreau proved that any twin…

When you come at the kings you best not miss

A tournament is an orientation of a complete graph. We say that a vertex $x$ in a tournament $\vec T$ controls another vertex $y$ if there exists a directed path of length at most two from $x$ to $y$. A vertex is called a king if it controls every vertex of the tournament. It is well known that every tournament has a king. We follow Shen, Sheng, and Wu (SIAM J. Comput., 2003) in investigating the query complexity of finding a king, that is, the number of arcs in $\vec T$ one has to know in order to surely identify at least one vertex as a king. The aforementioned authors showed that one always…

关于正则锦标赛中缺失的近生成平衡反定向树的注记

Note on near-spanning balanced antidirected trees missing from regular tournaments

本文构造了满足半度条件但无法嵌入高度对称锦标赛的近生成平衡反定向树，给出了Stein猜想在均匀次线性度解释下的反例，并确定了最优目标阶及最小宿主阶。

Chickens and Dukes

Following on the King Chicken Theorems originally proved by Maurer, we examine the idea of multiple flocks of chickens by bringing the chickens from tournaments to multipartite tournaments. As Kings have already been studied in multipartite settings, notably by Koh-Tan and Petrovic-Thomassen, we examine a new type of chicken more suited than Kings for these multipartite graphs: Dukes. We define an M-Duke to be a vertex from which any vertex in a different partite set is accessible by a directed path of length at most M. In analogy with Maurer's paper, we prove various structural results regard…

Subset-FAST的二次顶点核与次指数算法

A Quadratic Vertex Kernel and a Subexponential Algorithm for Subset-FAST

本文首次为Subset-FAST问题构造了二次顶点核，通过改进FAST归约规则并新增两条规则识别无关顶点，进而得到首个次指数时间FPT算法。

寻找国王与强国王的难度

Hardness of Finding Kings and Strong Kings

本文研究了有向图中寻找国王与强国王的查询复杂度，证明了在任意 $n$ 顶点有向图中确定国王存在性及在竞赛图中寻找强国王的查询复杂度均为 $Θ(n^2)$，并设计了特定竞赛图以辅助证明。

五个选民无法诱导的锦标赛

Tournaments not inducible by five voters

本研究通过定制搜索算法，将五个选民不可诱导锦标赛的最小阶数界限改进为13至23，并证明P23等实例不可诱导，同时给出可验证的机器证明。