Framed 4-valent graph minor theory I: Introduction. A planarity criterion and linkless embeddability

Abstract

The present paper is the first one in the sequence of papers about a simple class of 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 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.

A circle criterion for a generalized cross graph in terms of minimal excluded minors

“…In [3] and [4] Manturov proposed to consider an analogue of minor-closed properties for cross graphs (see [5] and [6]). Cross graphs are intimately related to both graphs of general form and knot theory.…”

“…The aim of this work is to develop a kind of generalization of the theory of minors to cross graphs; see [3] and [4]. Namely, with each cross graph we associate an equivalence class of framed graphs with respect to certain transformations; see the definitions below.…”

“…Definition 2.4. A 4-graph is called a cross graph, a graph with cross structure, a graph with the structure of opposite edges, or a framed 4-valent graph (see [3]- [6] and [8]) if for every vertex of it the four emanating half-edges are split into two pairs. The half-edges in one pair are called (formally) opposite to each other, while half-edges in different pairs are said to be adjacent.…”

“…Our aim is to give a description of all minors forbidden in a circle generalized cross graph, that is, to compose a list of graphs such that a generalized cross graph is a circle graph if and only if it contains no minors generated by any graph in the list; see [3] and [4]. Clearly, it suffices to describe the minimal excluded minors, that is, the graphs such that any of their minors on fewer vertices is a circle graph.…”