path-intersection graph modeling

The recognition algorithms for path graphs and directed path graphs have long remained disjoint, lacking a unified framework. Method: Leveraging a recent intersection graph characterization of path graphs, we extend this theory to the directed setting for the first time, establishing a shared structural property: both classes are precisely the intersection graphs of families of paths in a (directed) tree. Our algorithm avoids complex data structures, relying solely on tree decomposition and path cover verification. Contribution/Results: We present the first unified, linear-time verifiable recognition framework for both graph classes, achieving simultaneous recognition in $O(n^2)$ time. The approach is conceptually simple, rigorously verifiable, and yields a novel paradigm for designing uniform algorithms for interval graph subclasses and related families.

This work addresses path-based hypergraph visualization, modeling hyperedges as paths supported on underlying structures—including trees, cacti, and planar graphs with maximum degree four—and simultaneously optimizes three bend metrics: (i) minimizing total bends, (ii) minimizing the maximum bends per path, and (iii) maximizing, by layer, the number of zero-bend, one-bend, etc., paths. We introduce the first systematic multi-objective bend-optimization framework, unifying straight-line drawings (for trees and cacti) and orthogonal drawings (for planar 4-regular graphs), and integrate combinatorial optimization with geometric constraints. Experiments demonstrate that our method significantly reduces bend complexity across diverse structures, enhancing readability and visual quality of hypergraphs rendered as metro maps. The approach provides both theoretical foundations and practical algorithms for automated layout of path-supported hypergraphs.

This paper studies robust subexponential algorithms for the Hamiltonian Path/Cycle and Parameterized Long Path problems on geometric intersection graphs of similarly sized “fat” objects in ℝᵈ. We introduce the novel tool of λ-connected partitions and establish a low-treewidth pattern-covering theorem that is independent of concrete geometric representations, unifying divide-and-conquer with structural graph analysis. Our approach yields the first ETH-tight robust algorithms: Hamiltonian Path/Cycle is solved in 2ᴼ(ⁿ¹⁻¹⁄ᵈ) time, and the k-Path problem in 2ᴼ(ᵏ¹⁻¹⁄ᵈlog²k) ⋅ nᴼ(¹) time. These results overcome longstanding geometric-dependence barriers, significantly advancing the theoretical frontier of parameterized algorithms on geometric graphs.

This paper investigates geometric representations of hypergraphs: mapping vertices to points in the plane and hyperedges to curves (lines, line segments, or polygonal chains with bends) connecting their incident vertices. It systematically studies two fundamental constraints—crossing (allowing or forbidding intersections between edges sharing no vertex) and overlapping (allowing or forbidding collinear edge segments). The authors establish, for the first time, that six classical variants are ∃ℝ-complete, thereby characterizing their intrinsic computational hardness. They correct and generalize a long-standing erroneous claim in the literature. A generalized model incorporating bent lines is introduced, accompanied by a rigorous representation-theoretic framework. For restricted input classes, polynomial-time recognition algorithms are devised, and critical counterexamples are constructed. Integrating techniques from computational geometry, real-number complexity theory, and combinatorial embedding theory, the work unifies the characterization of representability boundaries for hypergraph geometric visualization.

This work addresses the vertex and edge connectivity determination problem for embedded graphs with bounded crossings—such as $k$-planar graphs and $d$-map graphs. Moving beyond classical algorithms restricted to planar graphs, we propose the first unified parameterized algorithmic framework applicable to crossing-restricted graph classes. Our approach integrates embedding-driven unit covering analysis, path-constraint modeling, and parameterized dynamic programming, thereby lifting the planarity assumption. The algorithm achieves linear or near-linear time complexity, attaining theoretical optimality across multiple near-planar graph families. It is the first method capable of exact connectivity computation for optimal $k$-planar graphs and other non-planar embeddings. Moreover, it simultaneously handles both vertex and edge connectivity testing within a single framework, unifying previously disparate treatments.

This work addresses the high computational complexity of cut-set computation in multi-path ensemble attribute evaluation by proposing an efficient algorithm and developing a vectorized computing framework based on matrix operations, which reformulates path attribute calculations as parallelizable array operations. For the first time, this approach provides a practical implementation of the formal model for path set attributes, integrating an optimized cut-set algorithm with array-oriented programming languages to substantially improve computational efficiency. Empirical evaluations across network simulations of varying complexity demonstrate that the method yields predictable and acceptable execution times, thereby establishing a practical foundation for large-scale multi-path analysis.

Existing graph databases lack effective support for the tree-shaped substructures commonly found in property graphs. This work addresses this limitation by treating such tree substructures as first-class citizens and proposes a systematic management framework encompassing modeling, indexing, and query optimization. Drawing inspiration from XML structural indexing techniques, the approach enables efficient path queries within a relational graph database backend. Experimental evaluation demonstrates that the proposed method significantly improves path query performance, thereby validating the potential of structural indexing to enhance graph data management.

This study investigates the feasibility and computational complexity of subgraph reconfiguration via local operations while preserving specific graph properties. Introducing buffer space as a resource-augmentation mechanism, the work proposes the first resource-aware reconfiguration model. By leveraging structural parameters such as pathwidth, treewidth, and minor-closed graph classes, combined with complexity analysis and constructive algorithms, it establishes that the reconfiguration problem remains NP-hard both on connected graphs of bounded pathwidth and on general graphs. Furthermore, the study reveals stark differences in buffer requirements across graph classes: planar graphs and those with bounded pathwidth or treewidth necessitate Ω(n) additional buffer vertices, whereas cactus graphs under certain constraints admit reconfiguration with only O(1) buffer space.

This work proposes an end-to-end approach based on Satisfiability Modulo Theories (SMT) for automatically generating maze paths from arbitrary input patterns—such as text or shapes—that satisfy prescribed topological constraints, with support for both two-dimensional and three-dimensional physical realizations. The method formalizes path synthesis as a global constraint satisfaction problem encompassing adjacency, continuity, and pattern coverage, enabling the generation of self-avoiding planar paths or layered paths with specified over-and-under crossings through a single SMT solving step. To the best of our knowledge, this is the first application of SMT-based formal methods to pattern-driven maze generation, unifying path synthesis and 2D/3D geometric construction. The framework successfully translates textual or graphical inputs into fabricable maze structures, validated through a complete pipeline and demonstrated via representative examples.

This study addresses the problem of reconfiguring a graph embedding on a surface—without introducing edge crossings—between two crossing-free embeddings that share fixed vertex positions, by means of edge rewiring. Integrating techniques from topological graph theory and combinatorial reconfiguration, the work establishes for the first time that matchings, trees, and forests are always reconfigurable on the torus and on any orientable surface of genus at least one. These results are further extended to the projective plane, thereby transcending the prior limitation to planar surfaces. The paper demonstrates the universal reconfigurability of several important graph classes on high-genus surfaces while simultaneously showing that general graphs are not always reconfigurable, thus precisely delineating the boundary of feasibility for this problem.