enumerate graph realizations

This paper studies the graph realization problem with subset cut constraints: given a nondecreasing degree sequence and prescribed edge-cut sizes across several vertex subsets, determine whether a simple graph satisfying both exists. We establish the precise computational complexity boundary by generalizing the Erdős–Gallai theorem, constructing polynomial-time reductions, and designing combinatorial algorithms. Specifically, the problem is solvable in $O(n^3)$ time when all cut constraints are imposed on subsets of size at most three; however, it becomes NP-complete as soon as 4-vertex subsets are allowed—even in the restricted case where all vertex degrees equal one. This sharp dichotomy fully resolves the critical threshold between tractability and intractability with respect to cut-constraint cardinality, thereby providing both a theoretical foundation and efficient algorithmic tools for constrained graph synthesis.

This paper establishes a rigorous categorical semantics for e-graphs (equivalence graphs) within the framework of monadic categories, supporting double-pushout (DPO) rewriting. Method: The authors generalize e-graphs to monadic categories by introducing *equivalence hypergraphs* (e-hypergraphs)—a compositional structure whose vertices are algebras over a monad and whose hyperedges encode algebraic operations, thereby internalizing structural equations up to isomorphism. The approach integrates category theory, semilattice-enriched categories, and hypergraph-based combinatorial modeling to yield a sound and complete semantic framework. Contribution/Results: The resulting framework provides an algebraic and monadic foundation for equivalence reasoning in e-graph–based program optimization, and extends the formal applicability of e-graphs to SMT solving and algebraic optimization—enabling principled, categorical treatment of equational rewriting beyond traditional graph-based methods.

Existing methods for verifying termination of graph transformation systems suffer from limited applicability and fail to uniformly handle diverse DPO variants. Method: This paper introduces the Generalized Weighted Type Graph (GWTG) framework, the first to extend weighted type graph techniques to arbitrary adhesive categories. GWTG supports not only classical DPO but also prominent variants—including SPO, AGREE, and NAC extensions—while preserving decidability of termination checking. Contribution/Results: By integrating categorical modeling, abstract interpretation, and graph rewriting semantics, GWTG enables fully automated termination verification for a substantially broader class of graph transformation systems. The approach significantly enhances expressiveness and generality without compromising theoretical soundness or computational tractability. This work establishes a more robust and scalable formal foundation for termination analysis in model-driven engineering.

The graph realization problem asks whether a given {0,1}-matrix is the path-incidence matrix of some generating forest—a characterization equivalent to recognizing network matrices, a fundamental subclass of totally unimodular (TU) matrices with critical applications in mixed-integer programming. Existing algorithms adopt column-wise incremental construction, suffering from limited submatrix recognition scope and ambiguity arising from multiple realizations. This paper introduces the first efficient row-wise incremental algorithm: it uniquely characterizes graphic matrices via SPQR trees to resolve realization ambiguity; designs data structures compatible with the Bixby–Wagner column-wise framework; and enables synergistic row-wise extension and column-wise computation for arbitrary submatrix detection. The method significantly improves decision efficiency—especially for matrices admitting multiple realizations—and provides a novel computational tool for leveraging TU structure in optimization.

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.

This work proposes a formalization of algorithms within an intensional computability framework and clarifies their relationship to implementations in computational models. Treating computational models as monoid actions on configuration spaces, programs are modeled as dynamical systems constrained by such actions. Algorithms are defined as finite directed graphs of partial maps over edge-labeled abstract data structures, explicitly separating control flow from data operations. By leveraging tools from category theory, dynamical systems theory, and graph theory, the approach constructs a rigorous semantic framework that, for the first time, treats algorithms as abstract specifications of computational behavior and precisely characterizes the structure-preserving implementation relation between programs and algorithms, thereby deepening our understanding of the nature of computation.

Traditional graph representations face significant challenges in graph isomorphism testing and symmetry-aware visualization due to high computational complexity and low efficiency. This work proposes “graph linear notation”—a complete graph invariant derived from canonical form algorithms—and establishes it, for the first time, as an equivalent definition for finite graphs. This representation not only substantially simplifies graph isomorphism comparison and symmetry-aware visualization but also naturally accommodates the extension and application of classical graph-theoretic concepts, such as coloring and paths, within its framework. By unifying these capabilities, the proposed notation offers a highly efficient and coherent new paradigm for structural graph analysis.

This study addresses the problem of characterizing graphic sequences and generating all their realizations. By introducing a “2-reduction” operation—subtracting one from each of two elements in the sequence—that preserves graphic equivalence, the authors develop a unified inductive reduction framework. This approach not only streamlines the reproducibility of classical characterization theorems such as Erdős–Gallai but also enables efficient construction of all graph realizations corresponding to a given sequence. Furthermore, it yields a novel equivalent characterization of graphic sequences, uncovering a deep connection between their intrinsic mathematical structure and underlying algorithmic mechanisms.

This study addresses the problem of efficiently extracting sequent calculus derivations from linear logic proof nets without altering their underlying graph structure. To this end, we propose a local coloring graph model based on half-edge coloring, which generalizes Yeo’s theorem to uniformly identify splitting vertices. Our approach integrates the Danos–Regnier correctness criterion, a cusp minimization lemma, and path analysis techniques, enabling modular reconstruction of sequent derivations directly from the original proof net. The method applies to linear logic systems featuring the mix rule and lacking multiplicative units. Beyond unifying and simplifying several existing graph-theoretic results, our work establishes that, in the absence of cusp cycles, a splitting vertex always exists, thereby guaranteeing an efficient sequentialization procedure.

This study investigates the enumeration complexity of minimal redundant sets and maximal irredundant sets in graph structures, with a focus on associated graph classes such as bipartite graphs, co-bipartite graphs, and split graphs. Through structural graph analysis and the design of enumeration algorithms, the work establishes for the first time that enumerating minimal redundant sets is computationally intractable on split graphs and co-bipartite graphs. Conversely, it proves that the problem is solvable in polynomial time on (C₃, C₅, C₆, C₈)-free graphs and strongly orderable graphs. These results delineate precise tractability boundaries across multiple graph classes, resolve a long-standing open question, and significantly extend the landscape of graph classes admitting efficient solutions.