完全区间竞赛图的重构。II
Reconstruction of complete interval tournaments. II
摘要：
设$a, \ b \ (b \geq a)$和$n \ (n \geq 2)$是非负整数，设$\mathcal{T}(a,b,n)$是这样的广义竞赛图的集合，其中每对不同的参与者至多与$b$相连，并且至少与$a$弧相连。在\cite{Ivanyi2009}中，我们给出了一个判定给定的非负整数序列$D = (d_1, d_2,..., d_n)$能否实现为$T \in \mathcal{T}(a,b,n)$的出度序列的充要条件.推广了\cite{Ivanyi2009}的结果，证明了对于任意非负整数序列$D$，存在$f$和$g$，使得某个元素$T \in \mathcal{T}(g,f,n)$的出度序列为$D$，对于任意具有相同出度序列$D$的$(a,b,n)$ -竞赛图$T'$，存在$a\leq g$和$b\geq f$ .我们提出了一个$Θ(n)$算法来确定$f$和$g$和一个$O(d_n n^2)$算法来构建相应的比赛$T$。
英文摘要：
Let $a, \ b \ (b \geq a)$ and $n \ (n \geq 2)$ be nonnegative integers and let $\mathcal{T}(a,b,n)$ be the set of such generalised tournaments, in which every pair of distinct players is connected at most with $b$, and at least with $a$ arcs. In \cite{Ivanyi2009} we gave a necessary and sufficient condition to decide whether a given sequence of nonnegative integers $D = (d_1, d_2,..., d_n)$ can be realized as the out-degree sequence of a $T \in \mathcal{T}(a,b,n)$. Extending the results of \cite{Ivanyi2009} we show that for any sequence of nonnegative integers $D$ there exist $f$ and $g$ such that some element $T \in \mathcal{T}(g,f,n)$ has $D$ as its out-degree sequence, and for any $(a,b,n)$-tournament $T'$ with the same out-degree sequence $D$ hold $a\leq g$ and $b\geq f$. We propose a $Θ(n)$ algorithm to determine $f$ and $g$ and an $O(d_n n^2)$ algorithm to construct a corresponding tournament $T$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
