Eulerian-minors and a concise recursive characterization of 4-regular planar graphs

TL;DR

By introducing the Eulerian-minor operation and its extension Eulerian-minor*, the paper provides a Wagner-type excluded-minor framework for Eulerian graphs, establishing that Eulerian graphs are precisely those with no Eulerian-minor equivalent to $K_2$ and that the relation is a well-quasi-ordering. It extends the framework to planarity, proving that Eulerian-minors* preserve planarity and giving excluded minors $K_5$ and $K_{3,3}^\prime$ for planar Eulerian graphs, with a parallel outer-planar characterization using $K_{2,3}^\prime$ and $K_{4}^\prime$. It then gives a concise recursive construction for all 4-regular planar graphs, reducing to the bouquet $B_2$ and free-loops via admissible demotions and contractions, and describing how to build simple 4-regular planar graphs from the Octahedron. These results unify Eulerian-graph structure under a minor-like theory and provide practical recursive tools for generation and recognition, with connections to knot-theoretic constructions.

Abstract

An Eulerian-minor of an Eulerian graph is obtained from an Eulerian subgraph of the Eulerian graph by contraction. The Eulerian-minor operation preserves Eulerian properties of graphs and moreover Eulerian graphs are well-quasi-ordered under Eulerian-minor relation. In this paper, we characterize Eulerian, planar and outer-planar Eulerian graphs by means of excluded Eulerian-minors, and provide a concise recursive characterization to 4-regular planar graphs.

Figures (7)

• Figure 1: An admissible demotion operation.
• Figure 2: $K_{3,3}^\prime$.
• Figure 3: All possible configurations of $G_4~(\ncong K_{3,3}^\prime)$ in the case $b_1b_2, b_1b_3,b_2b_3\notin E(G_4)$.
• Figure 4: Forbidden Eulerian-minors of outer-planar graphs.
• Figure 5: A local structure for seeking a peripheral cycle.

Theorems & Definitions (26)

• Theorem 2.1: O.Ve
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Remark 2.5
• Proposition 2.6
• Proposition 2.7
• Theorem 2.8
• proof