Basic embeddings into a product of graphs

Abstract

The notion of a basic embedding appeared in research motivated by Kolmogorov–Arnold's solution of Hilbert's 13th problem. Let K,X,Y be topological spaces. An embedding K⊂X×Y is called basic if for every continuous function f:K→ R there exist continuous functions g:X→ R , h:Y→ R such that f(x,y)=g(x)+h(y) for any point (x,y)∈K. Let T i be an i-od. Theorem. There exists only a finite number of `prohibited' subgraphs for basic embeddings into R×T n . Consequently, for a finite graph K there is an algorithm for checking whether K is basically embeddable into R×T n . Our theorem is a generalization of Skopenkov's description of graphs basically embeddable into R 2 , and our proofs is a (non-trivial) extension of that one.