Abstract
One of the most important and applied concepts in graph theory is to find the edge cover, vertex cover, and dominating sets with minimum cardinal also to find independence and matching sets with maximum cardinal and their polynomials. Although there exist some algorithms for finding some of them (Kuhn and Wattenhofer, 2003; and Mihelic and Robic, 2005), but in this paper we want to study all of these concepts from viewpoint linear and binary programming and we compute the coefficients of the polynomials by solving a system of linear equations with variables.
Highlights
All graphs in this note are simple, connected, finite, and undirected, though it is probable that some of the obtained results are extendable to general or directed graphs.Let G V, E be a simple and connected graph with |V | n and |E| m; the edge cover and edge dominating polynomials are of degree m, and the vertex cover and dominating polynomials are of degree n, in which coefficient of xk is the number of edge cover, edge dominating, vertex cover, and dominating sets with k elements, respectively
The independence and matching polynomials are at most of degree n such that coefficient of xk is the number of independence and matching sets with k elements, respectively, for some positive integer k
As previous notations we have the following theorem for obtaining the minimum size of edge cover set
Summary
All graphs in this note are simple, connected, finite, and undirected, though it is probable that some of the obtained results are extendable to general or directed graphs. Let G V, E be a simple and connected graph with |V | n and |E| m; the edge cover and edge dominating polynomials are of degree m, and the vertex cover and dominating polynomials are of degree n, in which coefficient of xk is the number of edge cover, edge dominating, vertex cover, and dominating sets with k elements, respectively. A set L ⊆ E is an edge cover if every vertex v ∈ V is incident to some edge of L. A set Q ⊆ V is a vertex cover if every edge e ∈ E has at least one endpoint in Q. We denote the Adjacency matrix by A and Incidence matrix by R, in which A that aij n×n such aij the numbers of edges with endpoints vi and vj , 1.2 and R rij n×m in which.
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