A Tutte-type canonical decomposition of 3- and 4-connected graphs

TL;DR

This work develops a Tutte-type canonical decomposition for 4-connected graphs by introducing tetra-separations (mixing vertices and edges with a degree and matching constraint) and a mixed-tree-decomposition that canonically represents all totally-nested tetra-separations. The authors classify the torsos that arise in the decomposition into four basic types: quasi-$5$-connected pieces, cycles of small torsos, generalised double-wheels, and thickened $K_{4,m}$ (including sprinkled variants), with a canonical $Y$-$\Delta$ operation to handle quasi-4-connected components. As corollaries, they obtain a canonical decomposition for 3-connected graphs into quasi-4-connected pieces, generalised wheels, and thickened $K_{3,m}$, and outline applications to vertex-transitive graphs and related algorithmic consequences. The framework unifies previous approaches (e.g., Tridecomp) while enabling explicit descriptions and canonicity that facilitate algorithmic use and structural insights into graph isomorphism, minor theory, and group-theoretic decompositions.

Abstract

We provide a unique decomposition of every 4-connected graph into parts that are either quasi-5-connected, cycles of triangle-torsos and 3-connected torsos on $\leq 5$ vertices, generalised double-wheels, or thickened $K_{4,m}$'s. The decomposition can be described in terms of a tree-decomposition but with edges allowed in the adhesion-sets. Our construction is explicit, canonical, and exhibits a defining property of the Tutte-decomposition. As a corollary, we obtain a new Tutte-type canonical decomposition of 3-connected graphs into parts that are either quasi-4-connected, generalised wheels or thickened $K_{3,m}$'s. This decomposition is similar yet different from the tri-separation decomposition. As an application of the decomposition for 4-connectivity, in a follow-up paper we obtain a new theorem characterising all vertex-transitive finite connected graphs as essentially quasi-5-connected or on a short explicit list of graphs.

Figures (40)

• Figure 1.1: A circular saw has vertex-set $\mathbb Z_n\times\mathbb Z_2$ with $n\gg k$ and every vertex $(v,0)$ sends edges to the $k$ vertices $(v,1),\ldots,(v+k-1,1)$. It is $k$-connected, and when $k\geqslant 3$ its neighbourhoods of vertices are precisely its $k$-separators (\ref{['SawWorks']}).
• Figure 1.2: $k$ is odd. Red: $\lfloor k/2\rfloor$-cliques. Blue: $\lceil k/2\rceil$-cliques. Grey indicates all possible edges.
• Figure 1.3: $k$ is even. Red: $(\tfrac{k}{2}-1)$-cliques. Blue: $\tfrac{k}{2}$-cliques. Grey indicates all possible edges. The central vertex is an apex.
• Figure 1.4: The totally-nested tetra-separations of a graph. Their separators are coloured.
• Figure 1.5: A generalised double-wheel

Theorems & Definitions (262)

• Theorem 1
• Remark 1.2
• Remark 1.3
• Corollary 2
• Conjecture 1.4
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Example 2.4
• Theorem 3.1: Technical strengthening of \ref{['MainDecomp']}