Abstract
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.
Highlights
A graph is a figure composed of vertices and edges and is usually used to describe a specific relationship between things; vertices are used to represent things, and an edge connecting two vertices is used to represent the relationship between the two things. e intuitive diagrammatic representation of relationships has created widespread use of graphs in modeling systems in physical science and engineering problems; any system involving a binary relation can be represented by a graph [1]
We present the main results of this paper, including theoretic part and algorithm part. eoretic results consist of matrix formulations of externally independent set and minimum externally independent set of graphs
A subset S of vertices is called a kernel of a graph if S is both an externally independent set and an internally stable set of the graph. e matrix formulation and algebraic algorithm of internally stable set of graphs are presented in [26]
Summary
A graph is a figure composed of vertices and edges and is usually used to describe a specific relationship between things; vertices are used to represent things, and an edge connecting two vertices is used to represent the relationship between the two things. e intuitive diagrammatic representation of relationships has created widespread use of graphs in modeling systems in physical science and engineering problems; any system involving a binary relation can be represented by a graph [1]. Yue and Yan [5] gave a matrix formulation of k-internally independent sets and k-maximum internally independent sets of graphs by proposing several new sufficient and necessary conditions of k-internally independent sets and k-maximum internally independent sets Based on these conditions, they further designed algebraic algorithms that can find all the k-internally independent sets and k-maximum internally independent sets of a given graph. Eoretically, the STP of matrices can be applied in most of engineering and science fields, especially for the problems that can be presented as a discrete mathematical model. Based on the STP of matrices, a matrix approach to investigate graph problems has been established recently.
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