An algebraic approach to sets defining minimal dominating sets of regular graphs

Abstract

Suppose that V = {1, . . . , n} is a non-empty set of n elements, S = {S 1 , . . . , S m } a non-empty set of m non-empty subsets of V . In this paper, by using some algebraic notions in commutative algebra, we investigate the question arises whether there exists an undirected finite simple graph G with V (G) = V , where S is the set whose elements are the minimal dominating sets of G.