Framed 4-valent graph minor theory II: Special minors and new examples

In the present paper, we proceed with the study of framed 4-graph minor theory initiated in [V. O. Manturov, Framed 4-valent graph minor theory I: Intoduction planarity criterion, arxiv: 1402.1564v1 [Math.Co]] and justify the planarity theorem for arbitrary framed 4-graphs; besides, we prove analogous results for embeddability in [Formula: see text].

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

This paper is the first one in the sequence of papers about a simple class of framed 4-graphs; the goal of this 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 analogs 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 new proof of Vassiliev's conjecture

In the paper [First-order invariants and cohomology of spaces of embeddings of self-intersecting curves in ℝn, Izv. Math. 69(5) (2005) 865–912] on finite type invariants of self-intersecting curves, Vassiliev conjectured a criterion of planarity of framed four-valent graphs, i.e. 4-graphs with an opposite edge structure at each vertex. The conjecture was proved by Manturov [A proof of Vassilievs conjecture on the planarity of singular links, Izv. Math. 69(5) (2005) 169–178]. We give here another proof of Vassiliev's planarity criterion of framed four-valent graphs (and more generally, (even) *-graphs), which is based on Pontryagin–Kuratowski theorem.