We consider the problem of embedding the vertices of a plane graph into a small (polynomial size) grid in the plane in such a way that the edges are straight, non-intersecting line segments and faces are convex polygons. We present a linear-time algorithm which, given an n-vertex 3connected plane graph G (with n 3), nds such a straight-line convex embedding of G into a (n 2) (n 2) grid.

This work proves a new result on the correct convergence of Min-Sum Loopy Belief Propagation (LBP) in an interpolation problem on a square grid graph. The focus is on the notion of local solutions, a numerical quantity attached to each site of the graph that can be used for obtaining MAP estimates. The main result is that over an N ×N grid graph with a one-run boundary configuration, the local ...

We study the Excluded Grid Theorem of Robertson and Seymour. This is a fundamental result in graph theory, that states that there is some function f : Z → Z, such that for all integers g > 0, every graph of treewidth at least f(g) contains the (g×g)-grid as a minor. Until recently, the best known upper bounds on f were super-exponential in g. A recent work of Chekuri and Chuzhoy provided the fi...

Hierarchical planar graph embedding (sometimes called level planar graphs) is widely recognized as a very important task in diverse fields of research and development. Given a proper hierarchical planar graph, we want to find a geometric position of every vertex (layout) in a straight-line grid drawing without any edge-intersection. An additional objective is to minimize the area of the rectang...

A visibility drawing of a plane graph G is a drawing of G where each vertex is drawn as a horizontal line segment and each edge is drawn as a vertical line segment such that the line segments use only grid points as their endpoints. The area of a visibility drawing is the area of the smallest rectangle on the grid which encloses the drawing. A minimum-area visibility drawing of a plane graph G ...

A set S of vertices in a graph G(V,E) is called a total dominating set if every vertex v ∈ V is adjacent to an element of S. The total domination number of a graph G denoted by γt(G) is the minimum cardinality of a total dominating set in G. Total domination subdivision number denoted by sdγt is the minimum number of edges that must be subdivided to increase the total domination number. Here we...

Let e be an edge of a median graph G which is contained in exactly one 4-cycle. Then it is proved that G n e is a median graph. The converse holds as well provided that e is not contained is a subgraph isomorphic to the 3-cube minus a vertex. These results are used to give several characterizations of median grid graphs. (Grid graphs are subgraphs of complete grids, i.e. of the Cartesian produc...

We present an adaptation of transfer matrix method for signed grids, grid-cylinders and grid-tori. We use this adaptation to count the number of satisfying assignments of Boolean Formulas in 2-CNF whose corresponding associated graph has such grid topologies.

The complexity of the maximum leaf spanning tree problem for grid graphs is currently unknown. We determine the maximum number of leaves in a grid graph with up to 4 rows and with 6 rows. Furthermore, we give some constructions of spanning trees of grid graphs with a large number of leaves.

An exact formula is given for the maximum number of edges in a graph that admits a three-dimensional grid-drawing contained in a given bounding box. The first universal lower bound on the volume of three-dimensional grid-drawings is obtained as a corollary. Our results generalise to the setting of multi-dimensional polyline grid-drawings.