Plane Strong Connectivity Augmentation

• Plane Strong Connectivity Augmentation (PSCA) is a framework for minimally augmenting planar graphs to achieve strong or 2-connectivity while preserving embeddings and orientations.
• The topic covers both directed and undirected variants, revealing contrasting computational complexities from NP-completeness to fixed-parameter tractable cases.
• PSCA leverages specialized methods such as geodesic shortcutting, dynamic programming, and gadget-based reductions to meet stringent connectivity and planarity conditions.

Plane Strong Connectivity Augmentation (PSCA) encompasses a family of fundamental algorithmic problems in planar graph theory, aiming to make a planar (typically embedded) graph or oriented digraph strongly connected by a minimal augmentation, subject to topological and geometric constraints. The problem's complexity, achievable bounds, and algorithmic methods vary significantly depending on whether the input is undirected or oriented, abstract, or equipped with a plane embedding, and depending on whether edge weights are considered.

1. Formal Definitions and Problem Settings

The central notion in PSCA is the augmentation of a planar or plane oriented graph to satisfy strong connectivity while preserving planarity and orientation. The most recent formalization is as follows:

Given a plane oriented graph $D = (V, A)$ with a fixed planar embedding (no antiparallel arcs), and integer $k$, the PSCA problem asks whether there exists an arc set $X \subseteq V \times V$, $|X| \leq k$, such that the digraph $D + X = (V, A \cup X)$ is strongly connected, remains plane (no crossings in the extended embedding), and remains oriented (still no antiparallel arcs) (Bessy et al., 19 Dec 2025).

For undirected graphs, the variant is often stated for planar straight-line graphs (PSLGs): For $G = (V, E)$, where vertices are in general position in $\mathbb{R}^2$ and edges are noncrossing segments, the aim is to find a set $E^+$ of noncrossing segments (lying entirely in the faces of $G$) so that $G' = (V, E \cup E^+)$ is 2-vertex-connected, with $\|E^+\|$ minimized. The definition admits analogous 2-edge-connectivity variants (Akitaya et al., 2016).

Table 1 summarizes the key input/output requirements in several canonical PSCA scenarios.

Model	Input graph	Allowed augmentation	Connectivity goal
Plane oriented (digraph)	$D$ plane oriented	Oriented arcs, plane	Strongly connected
PSLG (undirected)	$G$ PSLG	Noncrossing segments in faces	2-vertex (or edge) connected
Abstract planar	Planar/plane	Topological/embedding-respecting	$k$-connected

2. Computational Complexity and Hardness

The decision version—whether any valid (possibly unbounded-size) augmentation exists that achieves the desired connectivity while maintaining planarity and orientation—is NP-complete for plane oriented graphs. This result holds even without a cardinality constraint, and no $2^{o(\sqrt{n})}$ algorithm exists under ETH (Bessy et al., 19 Dec 2025), via a reduction from Planar 3-SAT that encodes logical consistency in strongly connected components and enforces it through planar gadgets (literal, variable, clause).

For the undirected PSLG variant, minimum-weight 2-connectivity augmentation is in $\mathsf{P}$ for connected input (see Section 3), but becomes NP-hard for disconnected graphs (Akitaya et al., 2016). In the context of higher connectivity, more general augmentation settings (for example, $c \to k$ augmentation with $2 \leq c < k \leq 5$) are also NP-complete for all planar, plane, or PSLG embeddings, with reductions utilizing block gadgets arranged in planar fashion (Akitaya et al., 1 Sep 2025).

3. Structural Results and Tight Bounds

3.1 Weight Bounds for Undirected PSLGs

A connected PSLG $G = (V, E)$, $|V| \geq 3$, admits a 2-connected PSLG augmentation by adding new (noncrossing) edges of total Euclidean length at most $2\|E\|$. This bound is tight: There exist instances for every $\epsilon > 0$ where the optimal ratio $\|E^+\| / \|E\| > 2 - \epsilon$ (Akitaya et al., 2016).

The constructive approach is based on the decomposition of the facial walks into convex chains, and insertion of geodesic (shortcut) paths within faces, ensuring that every original edge is "covered" at most twice and that new edges do not cross existing ones. This geodesic shortcutting is fundamental to achieving the bound and is proven using dual graphs on the convex chain covers of the facial structure.

3.2 Connectivity Thresholds in Planar Augmentation

No planar graph, regardless of augmentation, can be made more than 5-vertex-connected. This is a direct corollary of the degree bound implicit in Euler's formula: Every planar graph has a vertex of degree at most 5, so $\kappa(G) \leq 5$ (Akitaya et al., 1 Sep 2025). Necessary and sufficient edge counts for achieving a particular connectivity are derived via double-counting and planarity constraints.

For digraphs, the existence of a strongly connected augmentation is nontrivial and depends on the interplay of plane orientation and strong components' structure, unlike in general planar digraphs where unbudgeted augmentation is trivial.

4. Algorithmic Methods and Parameterized Complexity

4.1 FPT Algorithm for Budgeted Plane Oriented PSCA

For the budgeted PSCA problem ($|X| \leq k$), there exists a fixed-parameter tractable algorithm with runtime $2^{O(k)} n^{O(1)}$ (Bessy et al., 19 Dec 2025). The key methodological innovations are:

1. Supported solutions: Any solution can be "shifted" (face by face) to one where augmenting arcs appear at a bounded set of support angles per face.
2. Face classification: Each face is classified as "simple" (exactly two local terminals) or "alternating" (at least four local terminals, corresponding to complex strong component cuts on the face boundary).
3. Enumeration and branching: The number of possible supported completions for alternating faces is at most $2^{O(k)}$ (a bound on $\sum_{F \in AF}lt(F)$), so all can be enumerated.
4. Minimum Dijoin subroutine: After fixing augmentations on alternating faces, the remaining simple faces yield a structured instance equivalent to the Minimum Dijoin problem, solvable in polynomial time (Frank's algorithm), making core use of derandomization and universal sets.

This schema yields the first FPT algorithm for a connectivity augmentation problem subject to planarity constraints in the directed case.

4.2 Dynamic Programming for PSLGs

In the undirected PSLG setting with connected input, the minimal-weight 2-connectivity augmentation is computable in $O(n^4)$ time. The approach utilizes dynamic programming over facial walks, recursively solving for minimal chords to resolve cut-vertices, with cost functions reflecting geometric and embedding constraints (Akitaya et al., 2016).

4.3 Further Algorithmic Insights

For higher connectivity (e.g., 3-connected PSLGs), feasible augmentation is generally impossible, and even the existence problem is NP-hard. For triangulations, the problem of achieving 4-connectivity by edge flips (interpreted as simultaneous augmentations within planar embeddings) is NP-complete, but admits an EPTAS via Baker's shifting technique and dynamic programming on bounded treewidth induced subgraphs (Akitaya et al., 1 Sep 2025).

5. Connections to Broader Planar and Beyond-Planar Augmentation

The $c \to k$ connectivity augmentation framework generalizes PSCA. For fixed classes $\mathcal G_1 \subseteq \mathcal G_2$, given a $c$-connected $G \in \mathcal G_1$, the task is to find a minimal edge set $F$ ensuring $G' = (V, E \cup F) \in \mathcal G_2$ is $k$-connected (Akitaya et al., 1 Sep 2025). For planar/plane/PSLG constraints, augmentation is capped at $k = 5$. Extensions to $\ell$-planar (locally crossing-bounded) settings allow higher connectivity: Any $k$-connected graph can be drawn so all edges have at most $O(k^2)$ crossings, and vice versa; achieving $k$-connectivity necessitates permitting $\ell = \Omega(k^2)$ crossings per edge. Constructions using cluster partitions, clique-matching insertions, and extremal geometric constructions (e.g., circulants on convex point sets) realize these trade-offs tightly.

The minimum flip distance between planar triangulations, equivalently the minimum cardinality augmentation to 4-connected planar graphs, is NP-complete, yet admits efficient polynomial-time approximation schemes (Akitaya et al., 1 Sep 2025).

6. Open Problems and Further Directions

Several related questions remain unresolved or partially addressed.

• The complexity of PSCA on plane graphs where the embedding is not specified is open (Bessy et al., 19 Dec 2025).
• Weighted variants are not covered by the current FPT approach; weighted Strong Connectivity Augmentation admits FPT algorithms with a higher parameter dependence (Bessy et al., 19 Dec 2025).
• For undirected augmentation, the analogous “Plane 2-VCA” (plane biconnectivity) is NP-hard, but fixed-parameter tractability in the edge budget remains open (Bessy et al., 19 Dec 2025).
• Beyond-planar augmentation models, considering geometric, topological, or combinatorial relaxations of planarity, enable higher connectivities and introduce nuanced new trade-offs (Akitaya et al., 1 Sep 2025).
• The role of strong structural gadgets (variable, literal, clause) in NP-hardness reductions persists in related augmentation problems and underpins both clarity in proofs and limits of efficient computability (Bessy et al., 19 Dec 2025, Akitaya et al., 2016, Akitaya et al., 1 Sep 2025).

Prominent references for complete proofs, algorithmic schemes, and geometric constructions include the works of H. A. Akitaya et al. on minimum-weight PSLG augmentation (Akitaya et al., 2016), Akitaya et al. on the price of planar graph connectivity (Akitaya et al., 1 Sep 2025), and Bessy et al. on FPT algorithms and NP-hardness for plane oriented digraphs (Bessy et al., 19 Dec 2025).