partition refinement

This work investigates graphs that maximize the number of iterations required by the Colour Refinement algorithm—so-called “long-refinement graphs”—with a focus on classifying all such graphs of maximum degree at most 4 and arbitrary degree. The central problem is to characterize the existence, structural properties, and termination behavior of graphs requiring the maximal possible number of refinement steps. Using reverse engineering, long-refinement graphs are encoded as compact strings; structural analysis and closure under complementation are then leveraged to achieve the first complete classification of long-refinement graphs up to maximum degree 4: except for one sporadic case, all low-degree instances belong to known families. Moreover, it is rigorously proven that no graph exists whose vertices are distinguished only in the final iteration. These results confirm the tightness of the Kiefer–McKay upper bound and settle the classification problem, thereby revealing the intrinsic complexity boundary of Colour Refinement.

This paper addresses the challenge of preserving global structural properties during graph coarsening. We propose a novel coarsening framework grounded in Gromov–Wasserstein (GW) geometry, which explicitly models structural distortion induced by node-pair merging as the GW distance between local neighborhoods. Building on this formulation, we design two efficient algorithms: Greedy Pair Coarsening (GPC), which optimizes merges greedily, and k-means-enhanced GPC, which further enforces cluster-level consistency. We theoretically derive an error bound guaranteeing approximation to the optimal coarse graph. Extensive experiments on six large-scale graph datasets and downstream clustering tasks demonstrate that our method consistently outperforms state-of-the-art graph coarsening approaches in compression ratio, structural fidelity, and task performance—particularly under high compression ratios, where it exhibits superior robustness.

Canonical labeling of highly symmetric random circulant graphs—including their directed variants—remains challenging, as conventional combinatorial approaches such as color refinement fail on vertex-transitive graphs due to symmetry-induced indistinguishability. Method: This paper introduces a novel hybrid framework integrating color refinement with vertex individualization. Its core innovation lies in deriving a unique canonical label solely from the counts of walks of all lengths from each vertex to the individualized vertex. The method unifies Tinhofer’s canonicalization procedure, the 2-dimensional Weisfeiler–Leman algorithm, and walk-counting analysis to overcome refinement stagnation caused by automorphic symmetry. Results: Experiments demonstrate efficient canonical labeling for almost all random circulant (and circulant directed) graphs, along with construction of their canonical Cayley representations. This significantly advances the theoretical frontiers of graph isomorphism testing and encoding of symmetric graphs.

This paper studies the graph editing problem: given an undirected graph (G) and an integer (k), determine whether (G) can be transformed into a block graph or a strictly chordal graph using at most (k) edge additions, deletions, or both (with variants restricting to completion or deletion only). The authors develop the first polynomial-size vertex kernels for all five variants—block graph editing and deletion, and strictly chordal graph completion, deletion, and editing—achieving kernel sizes of (O(k^2)), (O(k^3)), and (O(k^4)), respectively. Their approach leverages structural characterizations—including forbidden induced subgraphs (e.g., diamonds, darts, gems), cut-vertices, and true twin relationships—combined with modular decomposition and safe contraction techniques. These kernels resolve long-standing open questions, as polynomial kernels were previously unknown for these problems. The results establish fixed-parameter tractability for all variants and provide tight preprocessing foundations for subsequent algorithm design.

The computational complexity of graph isomorphism testing remains unresolved, particularly for highly symmetric graphs whose adjacency matrices possess repeated eigenvalues—causing ambiguity in the solution space for conventional methods. This paper introduces a novel continuous optimization framework: it reformulates the discrete matching problem via orthogonal and doubly stochastic relaxations, and—crucially—systematically characterizes how eigenvalue multiplicity governs the geometric structure of the feasible solution space. Building on this insight, we propose a subspace constraint strategy that effectively suppresses spurious solutions induced by symmetry. Our algorithm employs the Frank–Wolfe method, integrating spectral matrix analysis with iterative projection-based optimization. Extensive evaluation on highly symmetric benchmarks—including strongly regular graphs, complete graphs, and the Petersen graph—demonstrates substantial improvements in both efficiency and robustness of isomorphism detection.

Existing hypergraph partitioning methods often become trapped in local optima, limiting partition quality. This work proposes ComPart, a novel framework that integrates community structure guidance during both the initial partitioning and uncoarsening phases. It is the first to comprehensively incorporate community detection throughout the entire uncoarsening process and extends the theory of local dense decomposition from graphs to hypergraphs to generate high-quality initial partitions. By synergistically combining diverse community detection techniques, hypergraph local dense decomposition, and a multilevel partitioning strategy, ComPart consistently outperforms state-of-the-art methods on standard benchmarks, achieving significantly improved partition quality.

This paper studies the Minimum Subgraph Completion problem: given a graph $G$ and a target graph class $mathcal{C}$, find a smallest vertex subset $S$ such that the completion of the induced subgraph $G[S]$ belongs to $mathcal{C}$. We establish, for the first time, a systematic framework for polynomial-time solvability of this problem. Our approach resolves several nontrivial transformations—including bipartite/co-bipartite/split graph interconversions, regular bipartite graphs to chordal graphs, forests to fixed degenerate graph classes, and disconnected/2-connected graph conversions. Methodologically, we integrate structural graph analysis, modular decomposition, matching theory, and degeneracy-order-based dynamic programming to design compact, scalable, problem-specific algorithms. Our results fill a fundamental theoretical gap in polynomial-time tractability for subgraph completion optimization and provide the first unified algorithmic paradigm for graph class transformation.

This study addresses the problem of finding a strong maximum independent set in hypergraphs—defined as a vertex set containing at most one vertex from each hyperedge—which arises in applications such as the construction of perfect minimal hash functions. The work introduces, for the first time, nine specialized data reduction rules grounded in structural properties of hypergraphs, serving as a preprocessing phase to substantially shrink instance sizes. Empirical evaluation demonstrates that this approach reduces instances to an average of 22% of their original size within just 6.76 seconds of preprocessing time. When integrated with the state-of-the-art exact solver, the method yields an average speedup of 3.84×, with peak acceleration reaching 53×, and successfully solves a previously intractable instance.

This study addresses the optimal vertex elimination problem—a core NP-complete challenge in algorithmic differentiation that has long lacked scalable exact solvers for evaluating heuristic performance. The work models the problem as a vertex elimination sequence on directed acyclic graphs and introduces a novel integer programming formulation capable of solving instances one to two orders of magnitude larger than previous approaches. It establishes the first approximate lower bounds for both minimum fill-in and minimum operation count variants and designs a parameterized approximation algorithm based on minimum s-t cuts. Additionally, the paper provides tight theoretical analyses for both forward and reverse modes of automatic differentiation. A new medium-scale graph benchmark with known optimal solutions is introduced, enabling empirical validation that demonstrates the strong practical performance of state-of-the-art heuristics and significantly advances both the tractable scale and theoretical understanding of the problem.

This work addresses the computational inefficiency of maximum common subgraph (MCS) solvers, which stems from the vast search space and inadequate handling of graph symmetries. We propose the first dual symmetry-breaking framework that simultaneously identifies modular symmetries in both the variable graph and the value graph based on local neighborhood structures. By leveraging equivalence class reasoning, our approach prunes isomorphic subtrees while preserving solution optimality. This method represents the first systematic integration of symmetry handling across both graphs, substantially enhancing pruning efficiency. Experimental results on standard MCS benchmarks demonstrate that our approach outperforms the current state-of-the-art RRSplit algorithm, solving more instances and significantly reducing both computation time and search space.