In this paper, the problem of formulating and finding externally independent sets of graphs is considered by using a newly developed STP method, called semitensor product of matrices. By introducing a characteristic value of a vertex subset of a graph and using the algebraic representation of pseudological functions, several necessary and sufficient conditions of matrix form are proposed to express the externally independent sets (EISs), minimum externally independent sets (MEISs), and kernels of graphs. Based on this, the concepts of EIS matrix, MEIS matrix, and kernel matrix are introduced. By these matrices’ complete characterization of these three structures of graphs, three algorithms are further designed which can find all these kinds of subsets of graphs mathematically. The results are finally applied to a WSN covering problem to demonstrate the correctness and effectiveness of the proposed results.

中文翻译：

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图的EIS的矩阵表述及其在WSN覆盖问题中的应用

在本文中，通过使用一种新开发的STP方法（称为矩阵的半张量积）来考虑制定和查找外部独立图集的问题。通过引入图的顶点子集的特征值并使用伪函数的代数表示，提出了几种必要和充分的矩阵形式条件来表示外部独立集（EIS），最小外部独立集（MEIS）和图的核。在此基础上，介绍了EIS矩阵，MEIS矩阵和核矩阵的概念。通过这些矩阵对图的这三种结构的完全刻画，进一步设计了三种算法，它们可以在数学上找到所有这些图子集。