Enumeration issues of independent sets in grid-graphs

The question is how widely studied mathematical theory in the twentieth and twenty-first century. To solve various problems, many algorithms have been developed for different types of graphs. The study of lattice graphs is one of the urgent topics for the development of science, both natural and computer. During this work, structural graphic elements will be considered, and we will use multimethod design research. First, we consider an independent set of vertices with step 1 in the framework of the well-known theory. Then we analyzed the methods for the total number of independent vertex sets. We need to adapt well-known algorithms for our task. In particular, we consider adaptation to determining the total number of independent sets of vertices.i(m,n) and calculating the total number of maximal independent sets of vertices β(m,n) for grid graphs G size of m × n.