Current Trends in Graph Theory and Computational Geometry

The recent developments in graph theory and computational geometry reveal a significant focus on advancing understanding and solving complex problems through innovative approaches. Key areas of interest include the characterization of linear layouts in ordered graphs, the computation of crossing numbers under various restrictions, the study of flexible graph realizations, and the exploration of reachability in Vector Addition Systems with States (VASS). Notably, there is a growing emphasis on parameterizing problems to enhance tractability and on identifying forbidden patterns that constrain graph properties. Additionally, the field is witnessing advancements in the analysis of geometric transformations, such as folding polyhedral metric graphs, and in understanding induced paths in graphs with specific constraints.

In the realm of ordered graphs, researchers are introducing new patterns to characterize mixed linear layouts, bridging the gap between stack and queue layouts. This work not only provides a deeper understanding of graph structures but also opens avenues for more efficient algorithms. Similarly, the computation of crossing numbers, a classic problem in computational geometry, is being reframed within a generalized framework that accommodates topological and geometric constraints, offering new insights into the parameterized tractability of these variants.

Flexible graph realizations continue to be a challenging area, with recent studies proving NP-completeness for specific classes of graphs and proposing faster algorithms for checking the existence of NAC-colorings. This progress is crucial for applications in rigidity theory and structural engineering.

The exploration of VASS reachability through geometric dimensions is another promising direction, revealing new complexities and establishing PSPACE-completeness for certain dimensions. This work underscores the importance of dimensional analysis in understanding system dynamics.

Finally, the characterization of long induced paths in graphs, particularly through the lens of forbidden patterns, is yielding polylogarithmic bounds and new insights into graph structure and connectivity. This research is pivotal for network analysis and optimization problems.

Noteworthy Papers

• Forbidden Patterns in Mixed Linear Layouts: Introduces 'thick patterns' to characterize mixed linear layouts, offering a significant advancement in understanding ordered graphs.
• Computing Crossing Numbers with Restrictions: Proposes a generalized framework for crossing number variants, addressing computational challenges with topological and geometric constraints.
• Flexible Realizations on Sparse Graphs: Demonstrates NP-completeness for NAC-coloring on sparse graphs and introduces faster algorithms for checking flexibility.
• Reachability in VASS: Establishes PSPACE-completeness for geometrically 1-dimensional and 2-dimensional VASS, extending the understanding of system reachability.
• Long Induced Paths and Forbidden Patterns: Characterizes forbidden patterns that force the existence of long induced paths, providing polylogarithmic bounds and new insights into graph connectivity.