A Hamiltonian path of a graph is a simple path which visits each vertex of the graph exactly once. The Hamiltonian path problem is to determine whether a graph contains a Hamiltonian path. A graph is called Hamiltonian connected if there exists a Hamiltonian path between any two distinct vertices. In this paper, we will study the Hamiltonian connectivity of rectangular supergrid graphs. Supergr...

Algorithms for discrete energy minimization play a fundamental role for low-level vision. Known techniques include graph cuts, belief propagation (BP) and recently introduced tree-reweighted message passing (TRW). So far, the standard benchmark for their comparison has been a 4-connected grid-graph arising in pixel-labelling stereo. This minimization problem, however, has been largely solved: r...

This paper investigates the following question: Given a grid φ, where φ is a proper subset of the integer 2D or 3D grid, which graphs admit straight-line crossing-free drawings with vertices located at (integral) grid points of φ? We characterize the trees that can be drawn on a strip, i.e., on a two-dimensional n × 2 grid. For arbitrary graphs we prove lower bounds for the height k of an n × k...

In this paper, we present algorithms to produce orthogonal drawings of arbitrary graphs. As opposed to most known algorithms, we do not restrict ourselves to graphs with maximum degree 4. The best previous result gave an (m ? 1) (m 2 + 1)-grid for graphs with n nodes and m edges. We present algorithms for two scenarios. In the static scenario, the graph is given completely in advance. We produc...

We study the problem of embedding a guest graph with minimum edge-congestion into a multidimensional grid with the same size as that of the guest graph. Based on a wellknown notion of graph separators, we show that an embedding with a smaller edgecongestion can be obtained if the guest graph has a smaller separator, and if the host grid has a higher but constant dimension. Specifically, we prov...

We present a simple and versatile formulation of grid-based graph representation problems as an integer linear program (ILP) and a corresponding SAT instance. In a grid-based representation vertices and edges correspond to axisparallel boxes on an underlying integer grid; boxes can be further constrained in their shapes and interactions by additional problem-specific constraints. We describe a ...

We investigate the s-t-connectivity problem for directed planar graphs, which is hard for L and is contained in NL but is not known to be complete. We show that this problem is logspace-reducible to its complement, and we show that the problem of searching graphs of genus 1 reduces to the planar case. We also consider a previously-studied subclass of planar graphs known as grid graphs. We show ...

In this thesis we will attempt to close the gap between fast grid-based method and -approximation methods that solve the Weighted Region problem. In order to compare both methods, we will define a formula to calculate the -value of grid-based methods and define the bounds of the -value for -approximation methods. Furthermore, we will improve on heuristics for A* grid methods for weighted region...

In this paper, we explore a new convention for drawing graphs, the (Manhattan-) geodesic drawing convention. It requires that edges are drawn as interior-disjoint monotone chains of axis-parallel line segments, that is, as geodesics with respect to the Manhattan metric. First, we show that geodesic embeddability on the grid is equivalent to 1-bend embeddability on the grid. For the latter quest...

The present paper studies solvers for partial differential equations that work on dynamically adaptive grids stemming from spacetrees. Due to the underlying tree formalism, such grids efficiently can be decomposed into connected grid regions (clusters) on-the-fly. A graph on those clusters classified according to their grid invariancy, workload, multi-core affinity, and further meta data repres...