A coarse block-cut tree theorem

Abstract: We prove a coarse analogue of the classic fact that every graph can be decomposed along its cut-vertices into $2$-connected components. Precisely, we prove that for every graph $G$ and a positive integer $d$, $G$ admits a tree decomposition whose adhesion sets have weak diameter at most $3d+2$ so that no two vertices $u,v$ lying in the same bag can be separated by a set of weak diameter at most $d$ whose distance from $u$ and $v$ is more than $d$. By the Coarse Menger's Theorem for two paths, this condition admits also a dual formulation, phrased in terms of the existence of two paths that are far from each other and connect the vicinity of $u$ with the vicinity of $v$.

• The paper proves that every connected graph has a tree-decomposition with adhesion sets of weak diameter at most 3d+2 and bags free from small, distant separators.
• The authors adapt Berger–Seymour BFS layering by treating compact blocks as coarse cut-vertices and wide blocks as coarse 2-connected components, yielding an O(n(n+m)) construction.
• The paper derives a dual scattered-path formulation, showing that coarsely 2-connected vertex pairs admit two well-separated paths, with quantitative losses tied to the coarse Menger constant 129.

Overview and main result

This paper proves a coarse analogue of the classical block-cut tree decomposition: the fact that every graph decomposes, along its cut-vertices, into a tree of 2-connected components. The setting is coarse graph theory, where graphs are viewed as metric spaces via their shortest-path metric and intersection/disjointness conditions from structural graph theory are replaced by proximity/separation conditions. The main theorem is as follows.

Theorem. For every positive integer $d$ and every connected graph $G$, there exists a tree-decomposition $(T,\beta)$ of $G$ such that:

• each adhesion set has weak diameter at most $3d+2$; and
• no two vertices $u,v$ in a common bag can be separated by a set $S\subseteq V(G)$ with $\mathrm{diam}_G(S)\leqslant d$ and $\mathrm{dist}_G(S,\{u,v\})>d+1$.

Moreover, such a tree-decomposition is computable in time $O(n(n+m))$, where $G$0 and $G$1 are the vertex and edge counts. The second condition is a natural "coarse 2-connectivity" requirement: bags cannot be split by small-diameter separators that stay far away from the vertices being separated. The authors note that a very similar statement was independently proven by Albrechtsen and Georgakopoulos; the result is thus not unique to this manuscript, though the proof technique and the dual formulation developed here are contributions of this paper.

The theorem adds a positive instance to a mixed record of coarse analogues of classical results: coarse Erdős–Pósa and Gallai theorems hold, while the full coarse Menger theorem fails (with counterexamples by Nguyen–Scott–Seymour) and the coarse Excluded Grid Theorem fails. The block-cut tree joins the list of structures that survive the passage to the coarse regime.

The BFS-layering construction

The proof builds on the BFS-layering construction introduced by Berger and Seymour in their characterization of graphs quasi-isometric to trees. Fix a root $G$2 and partition $G$3 into layers $G$4 by distance from $G$5. Within layer $G$6, two vertices lie in the same block if they are connected in the subgraph induced by layers $G$7. Two elementary observations drive everything:

• all edges from a block of layer $G$8 to layer $G$9 lead to the same parent block, which implies that the adjacency graph of blocks is a tree $(T,\beta)$0; and
• any vertex of a block is connected to any vertex of a descendant block by a path avoiding all earlier layers.

A block is called $(T,\beta)$1-compact if it has weak diameter at most $(T,\beta)$2, and $(T,\beta)$3-wide otherwise. The key conceptual observation is that compact blocks act as "coarse cut-vertices": they have bounded diameter, hence behave like small separators in the metric sense, while wide blocks play the role of coarse 2-connected components.

The coarse block-cut tree-decomposition

Given the layering $(T,\beta)$4 and parameter $(T,\beta)$5, the paper constructs the $(T,\beta)$6-coarse block-cut tree-decomposition $(T,\beta)$7 with three kinds of nodes:

Node type	Bag
Compact block $(T,\beta)$8	$(T,\beta)$9 itself
Edge between two compact blocks $G$0	$G$1
Component $G$2 of $G$3	union of blocks in $G$4 plus adjacent compact blocks

Here $G$5 denotes the set of $G$6-compact blocks. A straightforward verification shows this is a valid tree-decomposition whose adhesion sets are precisely the $G$7-compact blocks, hence have weak diameter at most $G$8. It is computable in $G$9 time via repeated breadth-first searches.

To establish the second condition of the main theorem, the paper applies the construction with $3d+2$0 and proves that no bag contains a pair separable by a small far-away separator. The argument splits on the position of the separator $3d+2$1 relative to the least common ancestor block $3d+2$2 of the blocks containing $3d+2$3 and $3d+2$4:

• If $3d+2$5 lies strictly above $3d+2$6 in the layering, both vertices connect to any $3d+2$7 within deep layers, avoiding $3d+2$8.
• Otherwise, two lemmas show that each of $3d+2$9 and $u,v$0 has a path to the root in $u,v$1: either $u,v$2 is too close to the top of the layering relative to the vertex's own layer, or the intermediate block $u,v$3 at layer $u,v$4 must be $u,v$5-wide (since it is internal on the $u,v$6–$u,v$7 path in $u,v$8), so one of its vertices lies at distance more than $u,v$9 from $S\subseteq V(G)$0 and provides an escape route.

This completes the proof. Note the constant $S\subseteq V(G)$1 arises from the interplay between the adhesion bound and the width threshold used to guarantee escape vertices; the paper does not claim these constants are tight.

Dual formulation via scattered paths

Since the coarse Menger theorem holds for two paths — with the quantitative form due to Albrechtsen et al., yielding a hitting ball of radius $S\subseteq V(G)$2 around every path when no two paths at distance $S\subseteq V(G)$3 exist — the cut condition admits a dual flow condition. The paper defines:

• the $S\subseteq V(G)$4-cut condition: some $S\subseteq V(G)$5 with $S\subseteq V(G)$6, $S\subseteq V(G)$7 intersects every $S\subseteq V(G)$8-$S\subseteq V(G)$9 path;
• the $\mathrm{diam}_G(S)\leqslant d$0-flow condition: either $\mathrm{diam}_G(S)\leqslant d$1, or there exist two paths $\mathrm{diam}_G(S)\leqslant d$2 at distance $\mathrm{diam}_G(S)\leqslant d$3 from each other, each meeting the radius-$\mathrm{diam}_G(S)\leqslant d$4 balls around $\mathrm{diam}_G(S)\leqslant d$5 and $\mathrm{diam}_G(S)\leqslant d$6.

A duality lemma shows these conditions are approximately complementary: the cut condition always precludes the matching flow condition, and conversely, assuming $\mathrm{diam}_G(S)\leqslant d$7, failure of the $\mathrm{diam}_G(S)\leqslant d$8-flow condition implies the $\mathrm{diam}_G(S)\leqslant d$9-cut condition. The forward direction is elementary; the converse applies the two-path coarse Menger theorem to the balls $\mathrm{dist}_G(S,\{u,v\})>d+1$0 and $\mathrm{dist}_G(S,\{u,v\})>d+1$1.

Combining this duality with the main theorem yields the dual corollary: for every $\mathrm{dist}_G(S,\{u,v\})>d+1$2 and connected $\mathrm{dist}_G(S,\{u,v\})>d+1$3, there is a tree-decomposition with adhesion sets of weak diameter at most $\mathrm{dist}_G(S,\{u,v\})>d+1$4 such that any two vertices $\mathrm{dist}_G(S,\{u,v\})>d+1$5 in a common bag either satisfy $\mathrm{dist}_G(S,\{u,v\})>d+1$6, or admit two paths at distance more than $\mathrm{dist}_G(S,\{u,v\})>d+1$7 from each other, each connecting the radius-$\mathrm{dist}_G(S,\{u,v\})>d+1$8 neighborhoods of $\mathrm{dist}_G(S,\{u,v\})>d+1$9 and $O(n(n+m))$0. In words, coarsely 2-connected pairs in a bag are witnessed by two well-separated paths — the coarse counterpart of two internally disjoint paths through a 2-connected component. The factor $O(n(n+m))$1 reflects the loss inherent in the known two-path Menger constant.

Limitations and open questions

Several caveats bear directly on the strength of the results. First, the constants are not optimized: the adhesion bound $O(n(n+m))$2, the separation distance $O(n(n+m))$3, and especially the Menger-derived factors $O(n(n+m))$4 and $O(n(n+m))$5 are artifacts of the available tools, and tightening them would require improving the two-path coarse Menger constant. Second, the decomposition depends on the choice of BFS root, and the paper does not address canonicality or quasi-isometry-invariance of the resulting tree. Third, the theorem concerns finite simple graphs; extensions to infinite or locally finite settings, where much of coarse graph theory operates, are not treated here. Finally, the paper leaves open how this coarse block-cut structure interacts with the broader program of coarse minor theory — in particular whether it can serve as a building block for a coarse analogue of the Kuratowski–Wagner theorem, which remains open in general.

Conclusion

The paper establishes that the block-cut tree decomposition lifts to the coarse setting with explicit constants: adhesion sets of weak diameter $O(n(n+m))$6 and bags free of small far-away separators, computable in near-linear-plus-quadratic time. The proof is a clean repurposing of the Berger–Seymour BFS-layering machinery, identifying compact blocks as coarse cut-vertices. The derived dual formulation, via two separated paths, connects the result to the coarse Menger theorem for two paths and quantifies the price paid by its constant. Together with the independent work of Albrechtsen and Georgakopoulos, this confirms that 2-connected-component structure is among the classical decomposition theorems that survive coarseification.