用一次子竞赛反转运算使竞赛图不可分解
Making a tournament indecomposable by one subtournament-reversal operation
摘要：
给定一个锦标赛$T$，$T$的模是$V(T)$的子集$M$，使得对于$x, y\in M$和$v\in V(T)\setminus M$，$(v,x)\in A(T)$当且仅当$(v,y)\in A(T)$。$T$的平凡模块是$\emptyset$、$\{u\}$ $(u\in V(T))$和$V(T)$。竞赛图$T$是不可分解的，如果它的所有模块都是平凡的;否则它是可分解的。设$T$是一个至少有5个顶点的竞赛图。在以前的文章中，作者证明了使$T$不可分解的最小弧数$δ(T)$满足$δ(T) \leq \left\lceil \frac{v(T)+1}{4} \right\rceil$，并且这个界是尖锐的，其中$v(T) = |V(T)|$是$T$的阶.本文证明了如果竞赛图$T$是偶数阶不可传递的，则通过反转$T$的子竞赛图的弧，可以使$T$不可分解。我们用$δ'(T)$表示这样的子锦标赛的最小尺寸。我们还证明了$δ(T) = \left\lceil \frac{δ'(T)}{2} \right\rceil$ .
英文摘要：
Given a tournament $T$, a module of $T$ is a subset $M$ of $V(T)$ such that for $x, y\in M$ and $v\in V(T)\setminus M$, $(v,x)\in A(T)$ if and only if $(v,y)\in A(T)$. The trivial modules of $T$ are $\emptyset$, $\{u\}$ $(u\in V(T))$ and $V(T)$. The tournament $T$ is indecomposable if all its modules are trivial; otherwise it is decomposable. Let $T$ be a tournament with at least five vertices. In a previous paper, the authors proved that the smallest number $δ(T)$ of arcs that must be reversed to make $T$ indecomposable satisfies $δ(T) \leq \left\lceil \frac{v(T)+1}{4} \right\rceil$, and this bound is sharp, where $v(T) = |V(T)|$ is the order of $T$. In this paper, we prove that if the tournament $T$ is not transitive of even order, then $T$ can be made indecomposable by reversing the arcs of a subtournament of $T$. We denote by $δ'(T)$ the smallest size of such a subtournament. We also prove that $δ(T) = \left\lceil \frac{δ'(T)}{2} \right\rceil$.
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