具有完美匹配的图的P-顶点问题
The P-vertex problem for graphs with perfect matchings
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中文总结 AI 辅助
本文将二分图的P-顶点问题结果推广到任意图，引入P-顶点覆盖概念，通过极大匹配划分顶点集构造矩阵，借助隐函数定理完成证明。
中文摘要 AI 辅助
Sharma和Panda近期证明，每个具有完美匹配的二分图都满足性质(P)，即它存在一个支撑图为G的非奇异实对称矩阵，使得每个顶点都是P-顶点。本文将他们的结果从二分图推广到任意图。为此，我们引入P-顶点覆盖的概念，并定义P-顶点覆盖数p(G)为S(G)中所需的非奇异矩阵的最小数量，使得G的每个顶点至少是其中一个矩阵的P-顶点。给定G的一个极大匹配，我们将顶点集划分为被该匹配饱和的顶点和其余顶点，后者必然构成独立集。随后我们构造单独的矩阵来覆盖这两类顶点。我们使用隐函数定理作为扰动工具来确立所需的结果。
英文摘要
Sharma and Panda recently proved that every bipartite graph with a perfect matching has property (P); that is, it admits a non-singular real symmetric matrix with support graph G for which every vertex is a P -vertex. In this paper, we extend their result from bipartite graphs to arbitrary graphs. To this end, we introduce the notion of a P - vertex covering and define the P -vertex covering number p(G) as the minimum number of non-singular matrices in S(G) needed so that every vertex of G is a P -vertex of at least one of them. Given a maximal matching of G, we partition the vertex set into the vertices saturated by the matching and the remaining vertices, which necessarily form an independent set. We then construct separate matrices covering these two classes of vertices. We use the Implicit Function Theorem as a perturbation tool to establish the desired result.
