Basic embeddings into a product of graphs

The notion of a basic embedding appeared in research motivated by Kolmogorov–Arnoldʹs solution of Hilbertʹs 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 Ti be an i-od. m. There exists only a finite number of `prohibitedʹ subgraphs for basic embeddings into R×Tn. Consequently, for a finite graph K there is an algorithm for checking whether K is basically embeddable into R×Tn. Our theorem is a generalization of Skopenkovʹs description of graphs basically embeddable into R2, and our proofs is a (non-trivial) extension of that one.