Grid-Obstacle Representations with Connections to Staircase Guarding

In this paper, we study grid-obstacle representations of graphs where we assign grid-points to vertices and define obstacles such that an edge exists if and only if an $xy$-monotone grid path connects the two endpoints without hitting an obstacle or another vertex. It was previously argued that all planar graphs have a grid-obstacle representation in 2D, and all graphs have a grid-obstacle representation in 3D. In this paper, we show that such constructions are possible with significantly smaller grid-size than previously achieved. Then we study the variant where vertices are not blocking, and show that then grid-obstacle representations exist for bipartite graphs. The latter has applications in so-called staircase guarding of orthogonal polygons; using our grid-obstacle representations, we show that staircase guarding is \textsc{NP}-hard in 2D.Comment: To appear in the proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Guarding art galleries by guarding witnesses

Let P be a simple polygon. We de ne a witness set W to be a set of points su h that if any (prospective) guard set G guards W, then it is guaranteed that G guards P . We show that not all polygons admit a nite witness set. If a fi nite minimal witness set exists, then it cannot contain any witness in the interior of P ; all witnesses must lie on the boundary of P , and there an be at most one witness in the interior of any edge. We give an algorithm to compute a minimal witness set for P in O(n2 log n) time, if such a set exists, or to report the non-existence within the same time bounds. We also outline an algorithm that uses a witness set for P to test whether a (prospective) guard set sees all points in P

Most vital segment barriers

We study continuous analogues of "vitality" for discrete network flows/paths, and consider problems related to placing segment barriers that have highest impact on a flow/path in a polygonal domain. This extends the graph-theoretic notion of "most vital arcs" for flows/paths to geometric environments. We give hardness results and efficient algorithms for various versions of the problem, (almost) completely separating hard and polynomially-solvable cases

Quantum dynamics of molecules in 4He nano-droplets: Microscopic Superfluidity

High resolution spectroscopy of doped molecules in 4He nano-droplets and clusters gives a signature of superfluidity in microscopic system, termed as microscopic superfluidity. Ro-vibrational spectrum of 4HeN-M clusters is studied with the help of some important observations, revealed from experiments (viz., localised and orderly arrangement of 4He atoms, although, being free to move in the order of their locations; individual 4He atoms can not be tagged as normal/ superfluid, etc.) and other factors (e.g., consideration that the 4He atoms which happen to fall in the plane of rotation of a molecule, render a equipotential ring and thus, do not take part in rotation; etc.) which effect the rotational and vibrational spectrum of the system. This helps us in successfully explaining the experimental findings which state that the rotational spectrum of clusters have sharp peaks (indicating that the molecule rotates like a free rotor) and moment of inertia and vibrational frequency shift have a non-trivial dependence on N

Recognizing Sharp Features of 2-D Shapes

We present an efficient algorithm for recognizing and extracting sharp-features from complex polygonal shapes. The algorithm executes in O(n²) time, where n is the number of vertices in the polygon. Sharp-feature extraction algorithms can be useful as a pre-processing step for measuring shape-similarity between polygonal shapes

Path Planning in O/1/infinity Weighted Regions with Applications

Path Planning in O/1/infinity Weighted Regions with Application