几类图的公平集
Fair Sets of Some Class of Graphs
摘要：
给定一个图的顶点的非空集合$S$，一个顶点相对于$S$的距离是该顶点到$S$的顶点的距离的最大值和最小值之差。具有最小顶点的顶点构成集合的公平中心。任何顶点集，如果是某个顶点集的公平中心，就称为公平集。本文证明了任意公平集的导出子图是连通的，在树和图块的情况下，所有公平集的导出子图都是连通的弦图类.给出了$K_{n}$，$K_{m,n}$，$K_{n}-e$，轮图，奇圈和对称偶图的公平集.本文还讨论了笛卡尔积图的公平集。
英文摘要：
Given a non empty set $S$ of vertices of a graph, the partiality of a vertex with respect to $S$ is the difference between maximum and minimum of the distances of the vertex to the vertices of $S$. The vertices with minimum partiality constitute the fair center of the set. Any vertex set which is the fair center of some set of vertices is called a fair set. In this paper we prove that the induced subgraph of any fair set is connected in the case of trees and characterise block graphs as the class of chordal graphs for which the induced subgraph of all fair sets are connected. The fair sets of $K_{n}$, $K_{m,n}$, $K_{n}-e$, wheel graphs, odd cycles and symmetric even graphs are identified. The fair sets of the Cartesian product graphs are also discussed.
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。
