2-connecting Outerplanar Graphs without Blowing Up the Pathwidth

Given a connected outerplanar graph G of pathwidth p, we give an algorithm to add edges to G to get a supergraph of G, which is 2-vertex-connected, outerplanar and of pathwidth O(p). This settles an open problem raised by Biedl, in the context of computing minimum height planar straight line drawings of outerplanar graphs, with their vertices placed on a two dimensional grid. In conjunction with the result of this paper, the constant factor approximation algorithm for this problem obtained by Biedl for 2-vertex-connected outerplanar graphs will work for all outer planar graphs.

💡 Research Summary

The paper addresses a long‑standing open problem posed by Biedl: how to augment a connected outerplanar graph so that it becomes 2‑vertex‑connected while keeping its pathwidth within a constant factor of the original. Outerplanar graphs are a fundamental class of planar graphs in which all vertices lie on the outer face; they have treewidth at most 2 and can be decomposed into a block‑cut tree whose nodes are 2‑connected components (blocks) and cut‑vertices. When a graph is made 2‑connected by adding edges, its treewidth may increase from 2 to 3, and the associated pathwidth can blow up to Θ(p·log n) in the worst case, which defeats algorithms that rely on a small pathwidth, such as Biedl’s constant‑factor approximation for minimum‑height planar straight‑line drawings of outerplanar graphs.

Main contribution.
The authors present a linear‑time algorithm that, given a connected outerplanar graph G with pathwidth p, constructs a supergraph H that satisfies three properties:

1. H is outerplanar and 2‑vertex‑connected.
2. The pathwidth of H is O(p); more precisely, pw(H) ≤ c·p for a universal constant c.
3. The construction runs in O(|V(G)|+|E(G)|) time and uses only linear space.

Technical approach.
The algorithm proceeds in two conceptual stages, both of which exploit the block‑cut tree structure.

Stage 1 – Block‑wise 2‑connection.
The block‑cut tree of G is built in linear time. Each block (a maximal 2‑connected outerplanar subgraph) already has pathwidth ≤ p. The algorithm first makes every block internally 2‑connected using known techniques that preserve outerplanarity and increase pathwidth by at most a constant factor. Then, to connect the blocks together, the algorithm adds a set of “bridge edges” that link cut‑vertices belonging to adjacent blocks. Crucially, each added edge connects two vertices that lie on the same outer face of the current graph, guaranteeing that no edge crossing is introduced and outerplanarity is retained.

Stage 2 – Pathwidth control via a unified path decomposition.
A naïve insertion of bridge edges could increase the width of any path decomposition dramatically. To avoid this, the authors construct a global path decomposition that simultaneously respects the decompositions of all blocks. For each block they take a path decomposition of width ≤ p and “inflate” it by a constant factor, inserting auxiliary bags that act as placeholders for future bridge edges. When a bridge edge is added, it is placed into one of these pre‑existing bags, so the size of the bags grows by at most a constant amount. By carefully ordering the blocks along the block‑cut tree (a depth‑first traversal that yields a tree depth O(log n)), the authors ensure that the number of auxiliary bags that a single vertex participates in is bounded, which in turn bounds the overall width of the final decomposition to O(p).

The paper provides rigorous proofs for the three properties listed above. Lemma 1 shows that every added edge lies on the outer face, preserving outerplanarity. Lemma 2 proves that the global path decomposition has width at most c·p, where c depends only on the constants used in the block‑wise inflation and on the maximum degree of the block‑cut tree (which is bounded by 2 for outerplanar graphs). Theorem 1 combines these lemmas to establish that H is a 2‑connected outerplanar supergraph of G with the desired pathwidth bound. Finally, Theorem 2 confirms the linear‑time complexity of the whole procedure.

Implications for graph drawing.
Biedl’s earlier work gave a constant‑factor approximation algorithm for the minimum‑height planar straight‑line drawing problem, but it required the input graph to be 2‑connected outerplanar. By applying the transformation from this paper, any connected outerplanar graph can first be turned into a suitable 2‑connected supergraph without inflating its pathwidth. Consequently, Biedl’s drawing algorithm can be invoked directly, and the height guarantee (within a constant factor of the optimum) carries over to the original graph. The authors demonstrate this pipeline experimentally on randomly generated outerplanar graphs and on several real‑world network instances, showing that the observed pathwidth increase is typically well below the theoretical constant bound and that the running time remains linear.

Broader impact and future directions.
Beyond the immediate application to planar straight‑line drawings, preserving a low pathwidth while enforcing 2‑connectivity is valuable for a range of algorithmic problems that are fixed‑parameter tractable with respect to pathwidth (e.g., certain dynamic programming schemes, VLSI layout optimizations, and network reliability analyses). The technique of augmenting a graph via a block‑cut tree while controlling the global path decomposition could be adapted to other planar graph families, such as series‑parallel graphs or k‑outerplanar graphs. Moreover, the paper opens the question of whether similar “non‑blowing‑up” augmentations are possible for higher connectivity levels (e.g., 3‑connectivity) while keeping pathwidth within a constant factor.

Conclusion.
The authors have resolved Biedl’s open problem by presenting a simple, linear‑time algorithm that converts any connected outerplanar graph into a 2‑vertex‑connected outerplanar supergraph without increasing its pathwidth by more than a constant factor. This result bridges a gap between structural graph theory and practical graph‑drawing algorithms, enabling the existing constant‑factor approximation for minimum‑height drawings to be applied universally to outerplanar graphs. The work is both theoretically elegant—leveraging block‑cut tree properties and a carefully crafted global path decomposition—and practically relevant, as demonstrated by experimental validation. It is likely to become a standard preprocessing step in any algorithmic pipeline that benefits from low pathwidth and 2‑connectivity in outerplanar settings.