PLanAR
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Counting Hamiltonian Paths in 3-Regular Planar Graphs
Combinatorics [Submitted on 5 Jun 2026] Title:Counting Hamiltonian Paths in 3-Regular Planar Graphs View PDF HTML (experimental)Abstract:We introduce two infinite families of 3-regular planar graphs. Both families are conceptual adversaries to the Pohl-Warnsdorf algorithm for finding Hamiltonians.
PlanarBench: Evaluating LLM Spatial Reasoning via Planar Graph Drawing
arXiv:2606.02010v1 Announce Type: new Abstract: PlanarBench tests whether LLMs can draw planar graphs as ASCII art given only an edge list -- a spatial reasoning task that resists memorization because edge order, edge orientation, and node labels are all permutable. We evaluate 91 models on the 199 simplest non-isomorphic connected planar graphs (2 - 7 vertices).
Planar Symmetric Pattern Generation
arXiv:2606.02073v1 Announce Type: new Abstract: Generating objects with specific symmetries is essential in various real-world scenarios. However, adapting existing 2D continuous representations to enforce planar group symmetry remains a challenge, as the transformation of non-reflective group elements may disrupt continuity. To overcome this limitation, we propose a symmetrization framework for arbitrary planar groups.
A Near-Optimal Offline Algorithm for Dynamic All-Pairs Shortest Paths in Planar Digraphs
arXiv:2606.01809v1 Announce Type: new Abstract: In the planar, dynamic All-Pairs Shortest Paths (APSP) problem, a planar, weighted digraph $G$ undergoes a sequence of edge weight updates and the goal is to maintain a data structure on $G$, that can quickly answer distance queries between any two vertices $x,y \in V(G)$. The currently best algorithms for this problem require $\tilde{O}(n^{2/3})$ worst-case update and query time, while conditional lower bounds show that either update or query...
PLanAR: Planning-Language-Grounded Agentic Reasoning for Robot Manipulation
arXiv:2602.01662v4 Announce Type: replace Abstract: Recent advances in vision-language models (VLMs) have enabled increasing progress in real-world robot manipulation. However, long-horizon manipulation in unstructured environments requires VLMs to reason about changing scene states, action constraints, and execution outcomes, which remains difficult with natural language reasoning alone. We present PLanAR, a planning-language-grounded robot agent framework for open-vocabulary, long-horizon...
Planar Perfect Matching Counting is as Hard as Determinants
Announce Type: new Abstract: In the 1960s, Fisher, Kasteleyn and Temperley designed an ingenious algorithm for computing the partition function of the dimer model, or equivalently, for counting perfect matchings in edge-weighted planar graphs (Philos. Mag. 1961; J. Mathematical Phys. 1963). This FKT algorithm later became the foundation for Valiant's holographic algorithms (FOCS 2004; SIAM J. Comput. 2008), which motivated the study of counting problems under the Holant framework. Combined...
An adaptive Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) method for solving the biharmonic equation over planar multi-patch geometries
new Abstract: We present a novel adaptive isogeometric method for solving the biharmonic equation over planar multi-patch domains with possibly extraordinary vertices, parametrized by analysis-suitable G^1 multi-patch geometries. The proposed technique relies on the concept of Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP), which enforces the required C^1-smoothness of the solution across a common edge of two neighboring patches by imposing appropriate continuity conditions...
MB-Loc: Multi-planar Bird's-eye-view Localization in outdoor LiDAR scenes
Announce Type: new Abstract: Global LiDAR localization is a fundamental task for autonomous navigation systems. Recent methods perform Scene Coordinate Regression (SCR) and achieve superior accuracy over Absolute Pose Regression (APR) solutions by predicting dense 3D world coordinates. However, SCR approaches introduce two major bottlenecks: severe computational inefficiency from processing raw 3D geometries and significant performance degradation under varying sensor viewpoints.
Multi-planar 2D-U-Net Segmentation of 3D-CT Abdominal Organs augmented by Spatial Occurrence Maps
arXiv:2606.07717v1 Announce Type: cross Abstract: This work proposes a lightweight 2D-U-Net-based framework for segmenting five abdominal organs in large field-of-view 3D CT scans. The method combines coarse-to-fine segmentation, predictions from multiple anatomical planes, and additional fuzzy 3D spatial maps that provide anatomical location cues to improve segmentation accuracy.
On The Computational Complexity of Minimum Aerial Photographs for Planar Region Coverage
Announce Type: replace Abstract: With the popularity of drone technologies, aerial photography has become prevalent in many daily scenarios such as environment monitoring, structure inspection, law enforcement etc. A central challenge in this domain is the efficient coverage of a target area with photographs that can entirely capture the region, while respecting constraints such as the image resolution, and limited number of pictures that can be taken. This work investigates the...