二部图的临界集
Critical Sets in Bipartite Graphs
摘要：
设G=（V，E）是一个图.一个集合S是独立的，如果S中没有两个顶点是相邻的，alpha（G）是最大独立集的大小，而core（G）是所有最大独立集的交集。数字d（X）= |X| -|N（X）|是集合X的差，d_{c}（G）=max{d（I）：I是独立集}称为G的临界差.如果d（X）= d_{c}（G），则集合X是临界的.对于图G，我们定义ker（G）为所有临界独立集的交，而diadem（G）为所有临界独立集的并。对于二分图G=（A，B，E），具有二分图{A，B}，Ore定义了对A的每个子集X，delta（X）=d（X），而delta_0（A）=max{delta（X）：X是A的子集}.类似地定义了delta_0（B）。本文证明了对任意二部图G=（A，B，E），下列断言成立：d_{c}（G）=delta_0（A）+delta_0（B）; ker（G）=core（G）;|克（G）|+|王冠|=2*alpha（G）。
英文摘要：
Let G=(V,E) be a graph. A set S is independent if no two vertices from S are adjacent, alpha(G) is the size of a maximum independent set, and core(G) is the intersection of all maximum independent sets. The number d(X)=|X|-|N(X)| is the difference of the set X, and d_{c}(G)=max{d(I):I is an independent set} is called the critical difference of G. A set X is critical if d(X)=d_{c}(G). For a graph G we define ker(G) as the intersection of all critical independent sets, while diadem(G) is the union of all critical independent sets. For a bipartite graph G=(A,B,E), with bipartition {A,B}, Ore defined delta(X)=d(X) for every subset X of A, while delta_0(A)=max{delta(X):X is a subset of A}. Similarly is defined delta_0(B). In this paper we prove that for every bipartite graph G=(A,B,E) the following assertions hold: d_{c}(G)=delta_0(A)+delta_0(B); ker(G)=core(G); |ker(G)|+|diadem(G)|=2*alpha(G).
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。
