论文标题
ABC（t） - 图：与连接和g $^$^$连接的中位数的图形中值过程的公理表征
ABC(T)-graphs: an axiomatic characterization of the median procedure in graphs with connected and G$^2$-connected medians
论文作者
论文摘要
中位功能是一个位置/共识函数，该功能将任何配置文件$π$（一个有限的顶点）映射到一组顶点，将距离总和从$π$最小化到顶点。中值函数满足几个简单的公理：匿名（a），s之间（b）和一致性（c）。 McMorris，Mulder，Novick and Powers（2015）将共识函数的ABC问题定义为表征图表的问题（称为ABC-Graphs），为之满足公理（a），（b）和（c）的唯一共识函数是中位功能。 在本文中，我们表明具有$ g^2 $连接的中位数（尤其是双方HELLY图）的模块化图是ABC图形。另一方面，在所有图中的中值功能（axioms（t）和（t $ _2 $））中的中位功能满足了一些简单的局部公理，这使我们能够表明所有具有连接中位数的图表（包括Helly图，中位数图，中位数图，Matroids的基础图，Matroids的基础图，甚至$δ$ -Matroids $ -Matroids $ abct $ abct $ - 以及benzens $ - 以及Benzen $ - 以及$ - 以及$ - 以及$ - 以及$ - 以及$ - 以及$ - 以及$ - 以及$ - 以及$ - 以及$ - 以及$ raphs $ - 以及benzen $ - 以及benzen $ caphs $ caphers $ caplient $ anderient。 McMorris等人（2015年）证明了满足配对属性的图形（在他们的论文中称为Intersecting Interval属性）是ABC图形。我们证明，配对属性的图形构成了双方HELLY图的适当子类，我们讨论了此类图的识别问题的复杂性状态。
The median function is a location/consensus function that maps any profile $π$ (a finite multiset of vertices) to the set of vertices that minimize the distance sum to vertices from $π$. The median function satisfies several simple axioms: Anonymity (A), Betweeness (B), and Consistency (C). McMorris, Mulder, Novick and Powers (2015) defined the ABC-problem for consensus functions on graphs as the problem of characterizing the graphs (called, ABC-graphs) for which the unique consensus function satisfying the axioms (A), (B), and (C) is the median function. In this paper, we show that modular graphs with $G^2$-connected medians (in particular, bipartite Helly graphs) are ABC-graphs. On the other hand, the addition of some simple local axioms satisfied by the median function in all graphs (axioms (T), and (T$_2$)) enables us to show that all graphs with connected median (comprising Helly graphs, median graphs, basis graphs of matroids and even $Δ$-matroids) are ABCT-graphs and that benzenoid graphs are ABCT$_2$-graphs. McMorris et al (2015) proved that the graphs satisfying the pairing property (called the intersecting-interval property in their paper) are ABC-graphs. We prove that graphs with the pairing property constitute a proper subclass of bipartite Helly graphs and we discuss the complexity status of the recognition problem of such graphs.
