Iterations of Multifunctions for Graph Theory: Bipartite Graphs and Filters

Abstract

We present the theory of multifunctions applied to graphs. Its interesting feature is that walks are recognized as iterations. We consider the graphs with arbitrary number of vertices which are determined by multifunctions. The mutually unique correspondence between graphs and multifunctions is proven. We explain that many facts of graph theory can be formulated in the language of multifunctions and as examples we give$\colon$ neighborhood, walk, independent set, clique, bipartiteness, connectedness, isolated vertices, graph metric, leaf. To simplify the proofs of our theorems, we introduce the concept of iterations of multifunctions. The new equivalent conditions for bipartite multifunctions including the K\"onig theorem and even iterations theorem are given. We prove that there exist filters and ideals in graph theory that are similar to those from the set theory. Finally, to illustrate these facts, we consider the multifunction of prime numbers.