中位数图的交叉图的中位数度
A median degree from crossing graphs of median graphs
AI总结：
本文研究以图$X$为交叉图的中位数图空间$\text{Cross}^{-1}(X)$，证明有限中位数图交叉图同构的条件，定义并计算图的中位数度，刻画具有最大中位数度的图。
AI中文摘要：
中位数图$M$的交叉图$\text{Cross}(M)$定义为：其顶点是$M$的所有$\text{Θ}$-类，边则连接两个交叉的$\text{Θ}$-类。已知任意图$X$都可作为某一中位数图的交叉图实现。本文中，我们启动对所有以$X$为交叉图的中位数图构成的空间$\text{Cross}^{-1}(X)$的研究。首先，我们证明两个有限中位数图的交叉图同构当且仅当其中一个可通过一系列称为“滑动（slidings）”的基本变换从另一个得到。接着，受$\text{Cross}^{-1}(X)$总是包含唯一最大度中位数图（即$X$的单形图）这一事实的驱动，我们将$X$的中位数度定义为$\text{Cross}^{-1}(X)$中中位数图的最小可能度。我们计算了一些图族的中位数度，并刻画了具有最大中位数度的图。
英文摘要：
The crossing graph $\mathrm{Cross}(M)$ of a median graph $M$ is defined as the graph whose vertices are the $Θ$-classes of $M$ and whose edges connect two $Θ$-classes whenever they cross. It is known that every graph $X$ can be realised as the crossing graph of some median graph. In this article, we initiate the study of the space $\mathrm{Cross}^{-1}(X)$ of all the median graphs with crossing graph $X$. First, we prove that two finite median graphs have isomorphic crossing graphs if and only if one can be obtained from the other by a sequence of elementary transformations we call slidings. Then, motivated by the fact that $\mathrm{Cross}^{-1}(X)$ always contains a single median graph of maximal degree, namely the simplex-graph of $X$, we introduce the median degree of $X$ as the smallest possible degree of a median graph in $\mathrm{Cross}^{-1}(X)$. We compute the median degree for some families of graphs and characterise the graphs with maximal median degree.
