Bounded Carving-Width Plane Graphs

• Bounded carving-width plane graphs are planar graphs with a carving decomposition that limits cut-size and maintains width properties during planarization.
• They exhibit robust closure properties, enabling efficient algorithmic testing for embedded immersions and finite obstruction characterizations.
• The refined decomposition techniques support recursive analysis, leading to optimal parameterized algorithms and deeper insights in topological graph theory.

A plane graph of bounded carving width is a planar embedding of a graph for which there exists a carving decomposition whose maximum cut-size is bounded above by a fixed parameter. This class is fundamental in structural graph theory and algorithmic graph theory, as it arises as the image of the planarization of a bounded-carving-width graph, and retains the same width property. Bounded carving-width plane graphs enjoy robust closure and algorithmic properties, and play a key role in well-quasi-order theorems for embedded graph minors and immersions.

1. Carving-Width: Definitions and Decomposition

For a plane or abstract undirected graph $G=(V,E)$, a carving decomposition is a pair $(T, \lambda)$, where $T$ is an unrooted tree in which every internal node has degree three, and $\lambda$ is a bijection from the leaves of $T$ to the set $V$. Removing an edge $e$ in $T$ produces subtrees $T_1$ and $T_2$, whose leaves correspond to disjoint subsets $V_1, V_2 \subset V$. The width of edge $e$ is the number of edges in $E$ between $V_1$ and $V_2$:

$\mathrm{cut\mbox{-}width}(e) = |\{ \{u,v\} \in E : u \in V_1, v \in V_2 \}|$

The width of the decomposition is the maximum of $\mathrm{cut\mbox{-}width}(e)$ over all $e$ in $T$. The carving width $\mathrm{cw}(G)$ is the minimum width over all carving decompositions. For plane graphs, the embedding is fixed, but the definition remains combinatorial (Eppstein, 2017, Lunel et al., 3 Dec 2025).

2. Planarization and Preserved Boundedness Under Planar Embedding

Planarization is the transformation of a nonplanar graph $G$ into a planar graph $D$ by replacing each crossing of two edges in a plane drawing of $G$ with a new vertex of degree four. If $G$ has carving width $w$, Eppstein proves that there exists a planarization $D$ with $|V(D)| = n + O(w^2 n)$, where $n=|V(G)|$, and $\mathrm{cw}(D) \leq w$. Thus, for fixed $w$, the planarization produces a plane graph of linear size and bounded carving width (Eppstein, 2017).

The construction is explicit: an optimal carving decomposition $(T, \lambda)$ of $G$ is planar-embedded, graph edges are routed along the thickened tree, crossings are minimized using sorted routing in the rectangles corresponding to tree edges, and dummy vertices are inserted at crossings. The refined tree incorporating these dummy vertices yields a carving decomposition of $D$ with width at most $w$.

3. Structure Theorems for Decompositions in Planar Context

Every planar graph admits a carving decomposition with favorable geometric and topological properties. For 2-vertex-connected graphs, there exists a bond & linked carving decomposition: each cut is displayed by a connected subgraph and the minimality of cuts along any path in the decomposition tree is globally respected. More generally, any plane graph admits a disc & linked decomposition, where each cut is realized as the boundary of a topological disk (Jordan curve), assembled from block decompositions and cut-vertex trees (Lunel et al., 3 Dec 2025).

These decompositions have all internal nodes of degree three, O(|G|) nodes, and can be chosen so that all leaves are labeled. They allow a fine-grained analysis of the interaction between the plane embedding and the carving structure, essential for topological arguments.

4. Well-Quasi-Ordering Under Embedded Immersion

A central theorem is that, for each fixed $k$, the class of plane graphs with carving width at most $k$ is a well-quasi-order under embedded immersion. That is, for any infinite sequence $G_1, G_2, \ldots$ of plane graphs with $\mathrm{cw}(G_i) \leq k$, there exist indices $i < j$ and an embedded immersion $G_i \leq_{\rm im} G_j$ (Lunel et al., 3 Dec 2025).

Embedded immersion generalizes classical immersion by requiring that all paths respect the planar embedding—formally, as ordered immersions where the circular orderings at vertices are preserved, and all immersion-paths are tangent (do not interleave). The proof leverages the structure of linked disc decompositions, associates "leaving subgraphs" to decomposition edges, and applies the Nash–Williams lemma on trees to eliminate infinite antichains.

This result implies that any immersion-closed subclass within the bounded carving-width plane graphs admits a finite obstruction set.

5. Closure Properties and Algorithmic Consequences

The class of bounded carving-width plane graphs is robust: it is closed under planarization (no increase in carving width), under embedded immersion, and under the strong variant of immersion where no path passes through incidental vertices (Eppstein, 2017, Lunel et al., 3 Dec 2025). This closure persists without appealing to large structure theorems, and all decomposition constructions and embedding arguments are achieved by local transformations and rebranchings.

Algorithmically, if a carving decomposition of width $w$ is provided (or computed using known FPT algorithms in $2^{O(w)} n$ time), the planarization process and associated carving decomposition refinement take time $O(n + m w + n w \log w)$, which is linear in $n$ for fixed $w$. Testing for embedded immersion of a fixed pattern $H$ in a host plane graph $G$ of carving width at most $k$ is FPT parameterized by $k, |H|$, solvable in time $f(k, |H|) |G|$ (Lunel et al., 3 Dec 2025).

6. Relationships with Other Width Parameters and Graph Classes

If $G$ has bounded treewidth $\tau$ and maximum degree $\Delta$, then $\mathrm{cw}(G) = O(\max\{\tau, \Delta\})$, ensuring the planarization produces a planar graph of bounded treewidth (Eppstein, 2017). For planar graphs, bounded carving width is strongly correlated to bounded treewidth in the presence of degree constraints, and strictly weaker for unrestricted graphs. A plausible implication is that carving width is an optimal parameter for simultaneous topological control under planarization and immersion closure.

A summary of comparison:

Parameter	Preserved under Planarization?	Bounded-size Planarization?
Treewidth ($w$)	No (can increase to $\Omega(n)$)	No
Carving width ($w$)	Yes	Yes

7. Consequences, Obstructions, and Further Directions

Bounded carving-width plane graphs enable finite obstruction characterization for immersion-closed subclasses, enhance the tractability of immersion and minor testing algorithms, and admit rich recursive decompositions compatible with embedded and combinatorial graph structure. These properties facilitate the extension of well-quasi-order results to wider embedding-respecting frameworks, such as the medial digraphs used in bounded branch-width cases, and contribute to the proof that embedded graph minors are a well-quasi-order on all plane graphs (Lunel et al., 3 Dec 2025).

These developments suggest that bounded carving-width plane graphs are a natural combinatorial and topological class for the study of width-preserving graph transformations, and form a foundational piece for further exploration of parameterized algorithms and structural topological graph theory.

References:

• "The Effect of Planarization on Width" (Eppstein, 2017)
• "Well-quasi-orders on embedded planar graphs" (Lunel et al., 3 Dec 2025)