Framed 4-valent Graph Minor Theory I: Intoduction. A Planarity Criterion and Linkless Embeddability

The present paper is the first one in the sequence of papers about a simple class of {\em framed $4$-graphs}; the goal of the present paper is to collect some well-known results on planarity and to reformulate them in the language of {\em minors}. The goal of the whole sequence is to prove analogues of the Robertson-Seymour-Thomas theorems for framed $4$-graphs: namely, we shall prove that many minor-closed properties are classified by finitely many excluded graphs. From many points of view, framed $4$-graphs are easier to consider than general graphs; on the other hand, framed $4$-graphs are closely related to many problems in graph theory.