Well-Quasi-Ordering Eulerian Digraphs Embeddable in Surfaces by Strong Immersion

Abstract

We prove that for every surface $Σ$, the class of Eulerian directed graphs that are Eulerian embeddable into $Σ$ (in particular they have degree at most $4$) is well-quasi-ordered by strong immersion. This result marks one of the most versatile directed graph classes (besides tournaments) for which we are aware of a positive well-quasi-ordering result regarding a well-studied graph relation. Our result implies that the class of bipartite circle graphs is well-quasi-ordered under the pivot-minor relation. Furthermore, this also yields two other interesting applications, namely, a polynomial-time algorithm for testing immersion closed properties of Eulerian-embeddable graphs into a fixed surface, and a characterisation of the Erdős-Pósa property for Eulerian digraphs of maximum degree four. Further, in order to prove the mentioned result, we prove that Eulerian digraphs of carving width bounded by some constant $k$ (which correspond to Eulerian digraphs with bounded treewidth and additionally bounded degree) are well-quasi-ordered by strong immersion. We actually prove a stronger result where we allow for vertices of the Eulerian digraphs to be labeled by elements of some well-quasi-order $Ω$. We complement these results with a proof that the class of Eulerian planar digraphs of treewidth at most $3$ is not well-quasi-ordered by strong immersion, noting that any antichain of bounded treewidth cannot have bounded degree.

Figures (11)

• Figure 1: An infinite antichain of Eulerian digraphs with respect to strong immersions.
• Figure 2: Euerlian and strongly planar embeddings. If the embedding of $u$ is strongly planar, two edge-disjoint paths can cross at $u$, i.e., there are two edge-disjoint paths going from $i_1$ to $o_2$ and from $i_2$ to $i_1$. This is impossible in a Eulerian embedding.
• Figure 3: Two $6 \times 6$ Euler-grids depicted in black with underlying undirected grids depicted in orange. The right-hand side depicts a $6 \times 6$-swirl.
• Figure 4: The gadget for immersing Eulerian digraphs into a router.
• Figure 5: Gadget for the high representativity case. The $C_i$ are represented by the black circles. The paths $F_{u_1, v}, ...$ are marked in orange.

Theorems & Definitions (373)

• Theorem 1.1
• Proof
• Remark
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.5
• Conjecture 1.6
• Conjecture 1.7
• Theorem 1.8