Graph Edge Re-Growth Algorithm

• Graph Edge Re-Growth is a procedure that employs local edge-slide moves and dynamic scheduling to reconstruct and balance edge connectivity while preserving key invariants.
• It restores inter-partition connectivity in large-scale graph learning by accurately recovering cut edges, thereby ensuring consistent one-hop neighborhood aggregation in GNNs.
• The algorithm also supports network connectivity augmentation and stochastic growth models, enabling efficient reconstruction of hidden graphs and enhancing overall network resilience.

A Graph Edge Re-Growth Algorithm is a formal procedural mechanism—broadly interpreted across several graph-theoretic contexts—that reconstructs, augments, or restores graph edge structure to achieve prescribed combinatorial, functional, or statistical properties. Such algorithms generalize across discrete reconfiguration by edge moves (slides), post-partition edge restoration in GNN-based workloads, growth schedule synthesis in dynamic graphs, and connectomic augmentation for higher connectivity or target reconstruction accuracy. The following sections delineate the primary algorithmic and theoretical frameworks for edge re-growth, as exemplified in edge-slide transformation (Baird et al., 2010), logical verification and GNN workloads (Thorat et al., 23 Nov 2025), connectivity augmentation (Kortsarz et al., 2015), scheduling of dynamic re-growth (Mertzios et al., 2021), and dynamic network modeling (Yao et al., 2015).

1. Edge-Slide Re-Growth: Configuration Prescription

The edge-slide-based Graph Edge Re-Growth Algorithm prescribes a finite transformation sequence between two simple, connected graphs $G = (V,E)$ and $H = (W,F)$ of equal order and size, via local edge-slide moves. An edge slide replaces one endpoint of an edge—given three vertices $x \sim y \sim z$, with $x \not\sim z$—by removing $xy$ and adding $xz$. Iterative sequences of such slides, including compositions ("shuffles") and neighbor-set interchanges, enact a recursion that at each stage:

1. Raises a chosen vertex's degree to the full $n-1$ by repeated slides, utilizing a spanning tree and bounded by $n-1$ operations.
2. Reduces its degree by sliding non-essential edges into the remainder of the graph while maintaining connectivity.
3. Interchanges neighbors by further bounded slide sequences to achieve structural isomorphism in prescribed neighborhoods.
4. Removes the matched vertex, recursing on the $n-1$ reduced problem.

Algorithmically, the complete prescription (see full pseudocode in (Baird et al., 2010)) executes in $O(n^2 + ne)$ time and preserves connectivity, cardinality, and the Euler characteristic throughout. Termination and correctness are guaranteed by recursive reduction and explicit realization of all primitive operations as slide-equivalent moves. This mechanism can also be used to regularize graphs by minimizing the degree-energy functional $\sum_{v} d(v)^2$ via slides—which always strictly decrease the sum—thus yielding balanced, almost-regular configurations.

2. Edge Re-Growth in Partitioned Graph Learning

In the context of large-scale graph partitioning for logic synthesis verification, the Graph Edge Re-Growth Algorithm is pivotal for restoring inter-partition connectivity lost during upfront partitioning with tools such as METIS. Given initial partitions $\{S_i\}$ of a directed graph $G = (V,E)$, the algorithm identifies "boundary nodes" $B_i$ adjacent to $S_i$ but outside it, then reconstructs the set of all original cross-partition ("cut") edges $\Delta E$, inserting the set $C_i$ of relevant inter-partition edges back into the augmented subgraph $G_i^+$.

This process guarantees that for each node $u \in S_i$, its one-hop neighbors in the original graph are all present in the subgraph, which ensures that node representation updates under one-hop (GraphSAGE-style) GNN aggregation are identical to the unpartitioned baseline. The algorithm is linear in total edge count, with $O(m)$ cumulative cost for boundary identification and edge restoration. Empirically, on core EDA workloads, the re-growth step recovers nearly all accuracy lost by partitioning (e.g., up to 12.6 percentage points on 32-bit Booth multipliers), while significantly reducing GPU memory footprint and maintaining subgraph processability (Thorat et al., 23 Nov 2025).

3. Augmentation for Connectivity Restoration

Edge re-growth is elemental in network connectivity augmentation, particularly in the Tree Augmentation Problem (TAP): augmenting a given tree $T = (V,E)$ by a minimum subset of admissible "links" $F \subseteq \mathcal L$, such that the resulting graph is 2-edge-connected. The 1.5-approximation algorithm operates via sequential "credit-counting" contractions:

• Assign combinatorial credits ("coupons" and "tickets") to matchings and unmatched leaves.
• Iteratively contract subtrees covered by minimal feasible augmenting sets.
• At each contraction, ensure that the added links are "paid for" by local credits, maintaining a global invariant bounding the final augmentation size by $1.5|F^*|$ for any optimum $F^*$.
• Steps (greedy locking-tree or semi-closed subtree contractions) leverage the identification of shadows, twin-links, and locked leaves as per formal combinatorial definitions.

The overall runtime remains polynomial, dominated by maximum matching computations and small subtree searches (Kortsarz et al., 2015).

4. Dynamic Graph Edge Re-Growth Schedules

The dynamic model of edge re-growth, as characterized by growth schedules, refers to the orchestrated generation of a target graph $G$ from a minimal initial seed using a sequence of slot-based operations: vertex creation, local edge activation (within bounded distance $d$), and optional removal of certain edges ("excess edges"). Trade-offs between schedule length ($k$) and excess edge usage ($\ell$) are central. For distance $d=2$, algorithms yield:

• Zero-excess ($\ell=0$) schedules of length $n-1$ via candidate-elimination.
• Sublinear-length ($k = \lceil \log n \rceil$) schedules at the cost of $\Omega(n)$ or higher excess edge counts.
• Polynomial time algorithms exploit matching and 2-SAT reductions for schedule feasibility testing and construction of valid re-growth orderings (see explicit pseudocode in (Mertzios et al., 2021)).

The underlying hardness results demonstrate the NP-completeness of optimizing these trade-offs and establish lower bounds for various graph classes.

5. Edge Re-Growth Models in Stochastic Network Growth

Recursive growing network models often couple deterministic attachment of motif-based subgraphs to selected "bound-edges" with randomized edge removal and addition among existing vertices. In model $N(t)$, new motif copies are affixed at each bound-edge, while in $N'(t)$, each prior edge is removed with fixed probability $p_r$ and each non-edge is added with $p_a$, independent of the deterministic structural growth.

Key features include:

• Explicit motif preservation at every iterative step via injective embeddings.
• Exponential expansion in edge and vertex count controlled by the motif structure and bound-edge propagation ($|B(t)| =(2r)^t N_{e,0}$).
• Identical edge-cumulative degree distributions ($P_{\mathrm{e\text{-}cum}}(k)$) for both deterministic and randomized models, yielding asymptotic power-law tails, demonstrating preserved "scale-free" behavior under randomized re-growth and edge churn (Yao et al., 2015).

Per-iteration runtime remains exponential in iteration count due to bound-edge multiplication, but the process is effectively parallelizable.

6. Reconstruction and Inverse Problems

Algorithmic edge re-growth underpins methodologies for reconstructing hidden graphs under local constraints, such as the determination of edge structure from oracle queries (distance, betweenness, or all-distances). The "Simple" re-growth algorithm achieves $\tilde O(n)$ queries w.h.p. on random $\Delta$-regular graphs by reconstructing candidate edges based on the indistinguishability of their distance signatures relative to a small landmark set, followed by direct validation. More extensive multi-phase Voronoi-cell algorithms attain subquadratic bounds for general bounded-degree graphs, but best-known worst-case complexity remains open (Mathieu et al., 2021).

These frameworks also yield structural results on the metric dimension of random regular graphs, with edge re-growth playing a central role in efficient, parallelizable reconstruction.

7. Extensions, Limitations, and Open Directions

Across all frameworks, edge re-growth is typically local and parallelizable, but global optimization (especially under connectivity or functional constraints) is often computationally challenging. Limitations of current methods include:

• For edge-slide methods, lack of subquadratic slide bounds and no planarity control (Baird et al., 2010).
• In large EDA graph workloads, memory usage is fundamentally limited by the size and degree distribution of partitions; tighter control via multi-hop re-growth or more sophisticated kernel-dispatch policies remains largely unexplored (Thorat et al., 23 Nov 2025).
• For dynamic growth schedules, hardness of optimal schedule construction and the trade-off between time (slot count) and structural costs (excess edges).
• In stochastic models, exponential growth in resource requirements makes deep exploration of the regime $t \gg 1$ computationally intensive.

Potential research routes include tighter complexity upper bounds, decentralized and geometric-constraint-aware re-growth scheduling, and explicit extension to weighted or directed graphs. The connections between edge re-growth, network resilience, symmetry breaking, and reconstruction accuracy underscore its centrality in both theoretical and applied graph science.