Forbidding Kuratowski Graphs as Immersions

The immersion relation is a partial ordering relation on graphs that is weaker than the topological minor relation in the sense that if a graph $G$ contains a graph $H$ as a topological minor, then it also contains it as an immersion but not vice versa. Kuratowski graphs, namely $K_{5}$ and $K_{3,3}$, give a precise characterization of planar graphs when excluded as topological minors. In this note we give a structural characterization of the graphs that exclude Kuratowski graphs as immersions. We prove that they can be constructed by applying consecutive $i$-edge-sums, for $i\leq 3$, starting from graphs that are planar sub-cubic or of branch-width at most 10.

💡 Research Summary

The paper investigates the class of graphs that do not contain the two Kuratowski graphs, $K_{5}$ and $K_{3,3}$, as immersions. An immersion of a graph $H$ into a graph $G$ is a mapping that assigns each vertex of $H$ to a distinct vertex of $G$ and each edge of $H$ to a pairwise edge‑disjoint path in $G$ whose endpoints are the images of the incident vertices. This relation is strictly weaker than the topological‑minor (or subdivision) relation: every topological minor is an immersion, but many immersions are not topological minors. Consequently, the classical Kuratowski theorem—characterising planar graphs by the exclusion of $K_{5}$ and $K_{3,3}$ as topological minors—does not directly translate to immersion exclusion.

The authors provide a complete structural characterisation of the immersion‑free class. Their main theorem states that any graph that excludes both $K_{5}$ and $K_{3,3}$ as immersions can be built from two elementary families using a limited gluing operation:

1. Base families
   a. Planar sub‑cubic graphs – graphs that are planar and have maximum degree at most three. Because each vertex has degree ≤ 3, no pair of vertices can be linked by four internally disjoint paths, a necessary condition for immersing $K_{5}$ or $K_{3,3}$.
   b. Graphs of branch‑width at most 10 – branch‑width is a width parameter dual to tree‑width, measuring how “tangled’’ a graph can be. Prior work (e.g., Robertson–Seymour’s graph‑minor theory) shows that an immersion of $K_{5}$ or $K_{3,3}$ forces branch‑width ≥ 11. Hence any graph with branch‑width ≤ 10 automatically avoids these immersions.

2. Gluing operation: $i$‑edge‑sum for $i\le 3$
   An $i$‑edge‑sum takes two graphs $G_{1}$ and $G_{2}$, selects $i$ distinct edges in each, and identifies the $i$ edges pairwise, merging their endpoints. When $i=1,2,3$ the operation is called a 1‑, 2‑, or 3‑edge‑sum. The crucial technical lemma proves that if both operands are immersion‑free for $K_{5}$ and $K_{3,3}$, then the resulting graph after an $i$‑edge‑sum with $i\le3$ remains immersion‑free. The proof hinges on the observation that any new immersion would have to use the identified edges to create the required crossing structure, but at most three edges cannot support the four‑edge connectivity needed for $K_{5}$ or the bipartite crossing pattern of $K_{3,3}$.

The paper proceeds in three logical parts:

• Part I – Base families are immersion‑free.
  For planar sub‑cubic graphs, the authors argue via degree constraints: an immersion of $K_{5}$ would need a vertex of degree at least 4, which does not exist. For $K_{3,3}$, the bipartite nature forces each side to have three vertices of degree at least 3, again impossible in a sub‑cubic setting. For the branch‑width family, they invoke known lower bounds: any immersion of a non‑planar Kuratowski graph raises the branch‑width to at least 11. Hence graphs with branch‑width ≤ 10 cannot contain such immersions.

• Part II – Preservation under $i$‑edge‑sums.
  The authors formalise the $i$‑edge‑sum operation and prove a closure property. They consider a hypothetical immersion of $K_{5}$ or $K_{3,3}$ in the summed graph and show that the immersion must be confined to one of the summands or must use the identified edges. In the latter case, the limited number of identified edges (≤ 3) forces a cut of size ≤ 3 separating the two sides, contradicting the edge‑connectivity required by the Kuratowski immersions (both need at least 4 edge‑disjoint paths between certain vertex pairs). This argument uses Menger’s theorem and the definition of immersion to rule out any crossing that would realise the forbidden minors.

• Part III – Converse: every immersion‑free graph decomposes accordingly.
  Starting from an arbitrary graph $G$ that excludes $K_{5}$ and $K_{3,3}$ as immersions, the authors apply a decomposition based on minimal edge‑cuts. If $G$ has branch‑width ≤ 10, it belongs to the second base family directly. Otherwise, they locate a minimal edge‑cut of size at most 3 (guaranteed by the immersion‑exclusion condition) and split $G$ along this cut. Each component is recursively examined; the recursion terminates because each split reduces the size of the graph or its branch‑width. The result is a tree‑like composition of base graphs glued together by 1‑, 2‑, or 3‑edge‑sums, exactly matching the constructive description of the main theorem.

The significance of this result is twofold. First, it provides a complete characterisation of the immersion‑free class analogous to Kuratowski’s theorem for topological minors, but with a fundamentally different structural description. Second, the theorem yields algorithmic consequences: testing whether a graph excludes $K_{5}$ and $K_{3,3}$ as immersions can be reduced to checking branch‑width (a fixed‑parameter tractable problem) and verifying sub‑cubic planarity, followed by a decomposition into small edge‑sums, which can be performed in polynomial time.

In summary, the paper establishes that the graphs that do not immerse the two Kuratowski graphs are precisely those that can be assembled from planar sub‑cubic pieces and low branch‑width pieces using only 1‑, 2‑, or 3‑edge‑sums. This bridges a gap between immersion theory and classical planarity, extending the elegance of Kuratowski’s forbidden‑minor characterisation to the weaker immersion ordering.