Connectivity-Preserving Extensions

• Connectivity-preserving extensions are methods for augmenting systems while retaining specific connectivity properties across various domains like graphs, matroids, and multi-agent networks.
• They employ techniques such as element splitting, control barrier functions, and combinatorial enumeration to ensure that key connectivity metrics remain intact during system modifications.
• This framework enables robust applications in distributed robotics, network control, geometric modeling, and algorithmic separation, offering both practical control strategies and theoretical guarantees.

A connectivity-preserving extension is a principled modification or augmentation of a system—graph, matroid, multi-agent network, algorithmic separation routine, or structural prediction method—that enlarges the system while guaranteeing the preservation of a specified notion of connectivity. The connectivity being preserved may be graph-theoretic (e.g., $k$-connectivity, edge or vertex connectivity), matroidal, algebraic, or tied to network communication topology, depending on the context. Connectivity-preserving extensions have emerged in multiple areas, including matroid and graph theory, distributed control and multi-agent systems, geometric modeling, algorithmic graph separation, and reconfigurable robotics.

1. Matroidal and Graph-Theoretic Connectivity-Preserving Extensions

In matroid theory, an explicit connectivity-preserving extension is the element splitting operation for $p$-matroids (vector matroids over the prime field $GF(p)$), as formalized in "A note on connectivity preserving splitting operation for matroids representable over $GF(p)$" (Malavadkar et al., 2020). Given a connected $p$-matroid $M$ on ground set $E$, the standard splitting operation $M \to M_{a,b}$ increases the rank by one but may fail to preserve connectivity. A single-element extension—appending a new element $z$ to form $M'_{a,b}$—yields a connectivity-preserving extension provided the splitting is non-trivial (i.e., at least one circuit in $M$ containing exactly one of $\{a,b\}$ is no longer a circuit in $M_{a,b}$).

The element splitting operation is characterized by the following properties:

• If $M$ is connected and the splitting $M \to M_{a,b}$ is non-trivial, then $M'_{a,b}$ is connected.
• The rank function of $M'_{a,b}$ is explicitly described in terms of that of $M$ and the presence of so-called $np$-circuits.
• Preservation of deeper structure: if $M$ is Eulerian and admits an $ep$-decomposition with respect to $\{a,b\}$, then $M'_{a,b}$ remains Eulerian; similarly, it preserves Hamiltonicity under suitable conditions.

In topological graph theory, a comprehensive classification of connectivity-preserving extensions is provided for local orientation-preserving symmetry-preserving operations (lopsp-operations) on embedded graphs (Camp, 2023). The principal result is a simple necessary and sufficient "edge-preserving" condition for a lopsp-operation $O$ to always preserve 3-connectivity: $O$ must not introduce a type-1 edge between the type-1 and type-0 special vertices of the patch. This ensures preservation of 3-connectivity even for embeddings of face-width 1 or 2. Truncation, Ambo, Chamfer, Leapfrog, and similar geometrical operations are all edge-preserving; only Dual and some pathologically constructed operations fail.

2. Connectivity-Preserving Extensions in Multi-Agent Control and Distributed Robotics

Multi-agent networks routinely require connectivity-preserving control strategies to prevent loss of communication links during coordinated area coverage, exploration, or formation maintenance. Several control-theoretic frameworks implement such extensions:

• In "Connectivity-Preserving Multi-Agent Area Coverage via Optimal-Transport-Based Density-Driven Optimal Control (D2OC)" (Lee et al., 23 Nov 2025), connectivity preservation is enforced by embedding a smooth, strictly convex connectivity penalty into the optimal control problem—specifically, the quadratic cost associated with 2-Wasserstein distance matching is augmented by a log-sum-exp penalty on anticipated inter-agent distances that would violate a communication range threshold. This mechanism guarantees that the system remains in a connected topology over time, without rigidly fixing the formation or neighbor structure.
• For distance-dependent communication, Lyapunov-set-invariance theory underpins a variety of protocols. In "Connectivity-Preserving Swarm Teleoperation With A Tree Network" (Yang et al., 2018) and "Bounded Connectivity-Preserving Coordination of Networked Euler-Lagrange Systems" (Yang et al., 2018), distributed gradient-based control laws with adaptive gains and injected damping are constructed to ensure that no link in the prescribed edge set is ever lost, regardless of disturbances, bounded actuation, or model uncertainty.
• Control barrier functions (CBFs) have been deployed to formulate decentralized connectivity-preserving navigation for UAV swarms in the presence of obstacles and without explicit communication (Palani et al., 28 Nov 2024). Here, desired acceleration commands are subject to hard CBF constraints that enforce collision avoidance, connectivity edge length, and line-of-sight preservation, resulting in real-time control laws that provably maintain connectivity.

3. Algorithmic and Combinatorial Extensions: Connectivity-Preserving Separators

In parameterized algorithmics, connectivity-preserving extensions arise in separation problems where the cut set must guarantee not only separation between designated terminals but also preservation of intragroup connectivity or arbitrary monotone connectivity constraints. In "Connectivity-Preserving Important Separators: Enumeration and an Improved FPT Algorithm for Node Multiway Cut-Uncut" (Kenig, 19 Nov 2025), the authors define a connectivity-preserving separator as a minimal $(A,B)$-separator $S$ such that $S$ disconnects $A$ from $B$ and the induced constraints (e.g., within-group connectivity, global reachability, or custom logical compositions of these) are still satisfied in $G-S$.

The paper supplies:

• Enumeration algorithms for all minimal connectivity-preserving important separators of size at most $k$ in $2^{O(k\log k)}$ time, establishing tight combinatorial bounds.
• An FPT algorithm for the Node Multiway Cut-Uncut (N-MWCU) problem that improves prior complexity from $2^{O(k^2\log k)}$ to $2^{O(k\log k)}$.

This reflects a broader paradigm where the extension of standard cut formulations is made connectivity-aware, yielding combinatorially robust tractability results relevant for cut-uncut and network design problems.

4. Transformational and Topological Connectivity-Preserving Extensions

The preservation of connectivity in discrete geometric or robotic systems during transformation processes is a fundamental challenge. In "On Efficient Connectivity-Preserving Transformations in a Grid" (Almethen et al., 2020), the authors analyze the minimal-move shape transformation problem for $n$ devices on a 2D grid:

• For Hamiltonian shapes, there exists a connectivity-preserving transformation from any initial connected configuration $S_I$ to a final $S_F$ in $O(n\log n)$ line moves, matching the lower bound for the unconstrained (connectivity-breaking) case.
• More generally, a universal $O(n\sqrt{n})$ connectivity-preserving transformation exists for arbitrary shapes.
• The lower and upper bounds demonstrate sharp tradeoffs between the complexity of shape transformation and the requirement that connectivity is maintained at every step.

This model formalizes connectivity-preserving extension as the process of executing transformation sequences strictly within the class of connected configurations, with explicit complexity characterizations for worst-case scenarios.

5. Connectivity-Preserving Extensions in Structured Prediction and Geometry

In geometric learning and modeling tasks, explicit guarantees of topological fidelity are critical. "Auto-Connect: Connectivity-Preserving RigFormer with Direct Preference Optimization" (Guo et al., 13 Jun 2025) introduces a connectivity-preserving tokenization regimen for the prediction of tree-structured joint skeletons in automatic rigging:

• The tokenization scheme uses explicit markers (e.g., $\langle\mathrm{E1}\rangle$, $\langle\mathrm{E2}\rangle$) to encode hierarchical and parent–child relationships so that any token sequence produced by the model corresponds to a valid skeleton tree.
• A topology-aware reward function—incorporating both Chamfer distance (spatial) and connectivity-based metrics (tree edit distance and hierarchical Jaccard similarity)—enables fine-tuning via Direct Preference Optimization, steering the generative process toward anatomically sound and fully connected skeletons.
• The model's architecture ensures that disconnected skeletons or chains cannot be produced, as connectivity relationships are encoded directly at the sequence level.

This approach exemplifies a connectivity-preserving extension in the context of autoregressive modeling and inverse graphics.

6. Methodological Principles and Theoretical Guarantees

Across the above domains, connectivity-preserving extensions are founded on several methodological principles:

• Barrier or penalty construction: Hard or soft constraints (e.g., CBFs, barrier-type potentials, log-sum-exp relaxations) that encode connectivity requirements into the system's dynamics or optimization landscape.
• Topologically aware augmentation: Structural modifications that encode connectivity information and explicitly block the generation of disconnected configurations (as in tokenization or separator enumeration frameworks).
• Lyapunov-based invariance: Quantitative energy-like arguments ensure that the set of configurations with required connectivity constitutes a positively invariant set under the extension operation or controlled system.
• Combinatorial enumeration: Identification and controlled enumeration of combinatorial objects (separators, patches, circuits) that respect connectivity constraints, often leading to new parameterized or fixed-parameter algorithms.

These methodologies provide strong theoretical guarantees—often in the form of if-and-only-if theorems, explicit enumeration bounds, or invariance properties—underpinning the widespread adoption of connectivity-preserving extensions in maximally diverse technical domains.

7. Open Problems and Research Directions

Current research highlights several unresolved aspects:

• Complete combinatorial characterization of all possible split-pairs in matroids that yield nontrivial (connectivity-preserving) splittings (Malavadkar et al., 2020).
• Existence and design of universal connectivity-preserving transformation schemes in sub-$O(n\sqrt n)$ time for arbitrary reconfigurable grids (Almethen et al., 2020).
• Extensions of these frameworks to higher-dimensional geometric, algebraic, or hybrid network models, including higher $k$-connectivity or partial connectivity maintenance.
• Integration of connectivity-preserving extensions with data-driven methods in vision, graphics, or learning under novel structure compatibility constraints.

These open avenues underscore the continued impact and the central role of connectivity-preserving extension frameworks in systems engineering, combinatorics, optimization, and computational geometry.