construct graph decomposition

This paper addresses the fragmentation and lack of interoperability among structural complexity measures across graph theory, geometric group theory, and dynamical systems. Methodologically, it introduces a unified “structured decomposition” framework grounded in category theory: (i) it is the first to formalize diverse domain-specific decomposition paradigms categorically; (ii) it establishes a general duality theory linking decompositions to object completions; and (iii) it defines composable width functors enabling cross-model quantification, comparison, and translation of structural complexity. Key contributions include: a unified categorical characterization of over ten complexity parameters—including treewidth, layered treewidth, and hypergraph treewidth—revealing their intrinsic structural relationships; and a novel parameterized tractability paradigm for NP-hard problems, grounded in decomposition width. The framework achieves both theoretical unification and algorithmic realizability.

This paper addresses the efficient computation of the Tutte decomposition—a canonical tree decomposition into triconnected components—for 2-connected graphs. To handle fully nested 2-separations, we present the first structurally transparent, linear-time algorithm. Our method introduces the novel notion of “stability” to characterize nesting relations among 2-separations; integrates insights from the Cunningham–Edmonds theory with block-cut tree concepts and a recursive decomposition framework; and achieves an optimal $O(n+m)$ time bound via depth-first search coupled with dynamic structural maintenance. The algorithm attains the theoretical lower bound on time complexity while yielding a decomposition tree whose construction admits a clear combinatorial interpretation. This enhances both theoretical understandability and practical implementability, bridging a long-standing gap between conceptual elegance and computational efficiency in graph decomposition.

This work investigates the efficient decomposition of a graph’s edge set into a small number of complete bipartite subgraphs, with a focus on graph classes exhibiting low neighborhood complexity. By integrating Welzl’s vertex ordering with sublinear interval covering techniques, the authors develop a unified structural framework that transforms graphs satisfying the neighborhood interval property into compact biclique partitions. This approach not only generalizes several known upper bounds from combinatorics and algorithms but also yields near-tight bounds—within at most a logarithmic factor—for classical problems including the Zarankiewicz problem, matrix multiplication complexity, quantum circuit complexity, and structured shortest-path computation. The results substantially enhance the coherence and applicability of theoretical limits across these domains.

This work addresses efficient algorithms for bipartiteness-related graph problems—including $K_t$-subgraph covering, odd-cycle transversal, and maximum weighted cut—by introducing a novel graph width parameter: *bipartite treewidth* (btw), which unifies treewidth and odd-cycle transversal number. Building upon an enhanced tree decomposition framework, we develop the first dynamic programming paradigm tailored to btw, yielding fixed-parameter tractable (FPT) algorithms for multiple NP-hard problems. Furthermore, for any 2-connected graph $H$, we establish a complexity dichotomy theorem for $H$-odd-subgraph packing and related problems: under the btw parameterization, we precisely delineate the boundary between para-NP-completeness and XP-solvability. Integrating parameterized algorithmics, dynamic programming, and structural graph modeling, this work substantially extends the applicability of the treewidth framework to non-bipartite graph problems.

Structural characterization of 3-connected graphs remains a fundamental challenge in structural graph theory. Method: We establish the first canonical, unique, and explicit structural decomposition theorem for 3-connected graphs: every such graph admits a unique decomposition into three types of basic building blocks—quasi-4-connected graphs, wheels, or “thickened bipartite graphs” obtained by triangulating one side of $K_{3,m}$. This decomposition is constructive and invariant, overcoming the limitations of non-canonical, cutset-enumeration-based approaches. Contribution/Results: Leveraging this framework, we provide the first complete classification of Cayley graphs by vertex connectivity and derive a novel characterization theorem for them. Moreover, Tutte’s wheel theorem is automatically reconstructed and proved as a corollary. By unifying graph decomposition, canonical construction, group actions, and symmetry analysis, our work furnishes a unified structural framework and new algorithmic tools for graph algorithm design and automated theorem proving.

This work addresses the trade-off between low spread—where each vertex appears in few bags—and small width in tree decompositions, focusing specifically on domino tree decompositions (spread = 2). For graphs of treewidth $k$ and maximum degree $\Delta$, the authors develop a refined approach combining improved tree partitioning and chordal completion techniques, yielding decompositions whose underlying trees have maximum degree $O(\Delta)$ and $O(|V|/k\Delta)$ nodes, with tree partition width reduced to $O(k\Delta)$. The central contribution is a proof of an $\Omega(k\Delta^2)$ lower bound on domino treewidth, establishing the tightness of the known $O(k\Delta^2)$ upper bound and resolving an open question posed by Bodlaender regarding its optimality. Additionally, the paper shows that when spread is allowed to depend on $k$, a decomposition of width $O(k\Delta)$ becomes achievable.

This work investigates how to construct optimal-width tree decompositions while bounding the pathwidth of subgraphs induced by each bag. Focusing on planar graphs and graph classes excluding a fixed two-vertex forest as a minor, the study combines tools from graph decomposition theory, pathwidth and treewidth analysis, minor exclusion, and structural graph theory to prove that every planar graph admits an optimal tree decomposition in which each bag induces a subgraph of pathwidth at most three—a bound shown to be tight. The paper further provides an exact characterization of bag structures satisfying a minimality condition. These results extend to graphs embeddable on any fixed surface and yield a new proof of the linear grid minor theorem for planar graphs.

This study addresses an open problem posed by Blanco et al.: whether one can construct a tree decomposition whose index tree is a subgraph of the original graph and whose width is governed by pathwidth rather than treewidth. To this end, we present the first construction of such a tree decomposition for any connected graph, leveraging Simon’s Factorization Theorem. We rigorously prove that the width of the resulting decomposition is bounded above by a function of the graph’s pathwidth. This work not only establishes a theoretical connection between pathwidth and this class of subgraph-indexed tree decompositions but also transcends the conventional reliance on treewidth in tree decomposition theory, offering a novel perspective for structural graph analysis.

This work addresses the challenge of controlling the distribution of size-$k$ solution patterns—such as paths or packings—within tree decompositions of $H$-minor-free graphs, where traditional methods fail to effectively bound their presence in individual bags. The paper proposes a randomized polynomial-time algorithm that samples an induced subgraph together with a tree decomposition, which, with constant probability, fully preserves any given size-$k$ solution pattern while ensuring each bag contains only $\widetilde{O}(\sqrt{k})$ relevant vertices. This approach achieves the first simultaneous sparsification of both the solution pattern and its distance-$d$ neighborhood in $H$-minor-free graphs, overcoming prior limitations to connected patterns or planar graphs. By integrating graph minor theory, random sampling, and neighborhood control, the resulting structured decomposition enables randomized algorithms running in $2^{\widetilde{O}(\sqrt{k})}n^{O(1)}$ time for problems including Directed $k$-Path, $H$-Packing, and neighborhood-constrained variants such as Partial Dominating Set.

This study addresses the problem of connected dense partitioning in graphs, which seeks to maximize the density of each part under constraints on either the number of parts or their sizes. The authors propose a dynamic programming framework based on tree decompositions, yielding the first polynomial-time algorithm for this problem on thick forests and extending it to general graphs of bounded treewidth. Key contributions include an efficient method for constructing partitions that simultaneously satisfy connectivity and high-density requirements, a proof establishing the NP-hardness of the problem on split graphs, and an effective instantiation of the partition structure guaranteed by the Győri–Lovász theorem.