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The present paper is the first one in the sequence of papers about a simpleclass of {\em framed $4$-graphs}; the goal of the present paper is to collectsome 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 theRobertson-Seymour-Thomas theorems for framed $4$-graphs: namely, we shall provethat many minor-closed properties are classified by finitely many excludedgraphs.  From many points of view, framed $4$-graphs are easier to consider thangeneral graphs; on the other hand, framed $4$-graphs are closely related tomany problems in graph theory.

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