The bisection problem asks for a partition of the n vertices of a graph into two sets of size at most dn/2e, so that the number of edges connecting the two sets is minimised. A grid graph is a finite connected subgraph of the infinite two-dimensional grid. It is called solid if it has no holes. Papadimitriou and Sideri [7] gave an O(n) time algorithm to solve the bisection problem on solid grid...

Grid graphs are subgraphs of the infinite 2-dimensional integer grid. The Hamiltonian path problem for general grid graphs is a well-known NP-complete problem. In this paper, we present necessary and sufficient conditions for the existence of a Hamiltonian path between two given vertices in L−shaped grid graphs. We also show that a Hamiltonian path between two given vertices of a L−shaped grid ...

A grid drawing of a plane graph G is a drawing of G on the plane so that all vertices of G are put on plane grid points and all edges are drawn as straight line segments between their endpoints without any edge-intersection. In this paper we give a very simple algorithm to find a grid drawing of any given 4-connected plane graph G with four or more vertices on the outer face. The algorithm take...

We study the Hamiltonian Cycle problem in graphs induced by subsets of the vertices of the tiling of the plane with equilateral triangles. By analogy with grid graphs we call such graphs triangular grid graphs. Following the analogy, we define the class of solid triangular grid graphs. We prove that the Hamiltonian Cycle problem is NPcomplete for triangular grid graphs. We show that with the ex...

We present a short proof of the excluded grid theorem of Robertson and Seymour, the fact that a graph has no large grid minor if and only if it has small tree-width. We further propose a very simple obstruction to small tree-width inspired by that proof, showing that a graph has small tree-width if and only if it contains no large highly connected set of ver-tices.

We prove upper and lower bounds on the size of the largest square grid graph that is a subgraph, minor, or shallow minor of a graph in the form of a larger square grid from which a specified number of vertices have been deleted. Our bounds are tight to within constant factors. We also provide less-tight bounds on analogous problems for higher-dimensional grids.

A rectangular drawing is a plane drawing of a graph in which every face is a rectangle. Rectangular drawings have an application for floorplans, which may have a huge number of faces, so compact code to store the drawings is desired. The most compact code for rectangular drawings needs at most 4 f − 4 bits, where f is the number of inner faces of the drawing. The code stores only the graph stru...

The families of Edge Intersection Graphs of Paths in a tree (resp. in a grid) EPT (resp. EPG) are well studied graph classes. Recently we introduced the class of graphs of Edge-Intersecting and NonSplitting Paths in a Tree (ENPT) [2]. In this model, two vertices are adjacent if they represent two intersecting paths of a tree whose union is also a path. In this study we generalize this graph cla...

Some recent algorithms for 3-dimensional orthogonal graph drawing use no more than 3 bends per edge route. It is unknown if there exists a graph requiring a 3-bend edge route. In this paper we present an algorithm for 2-bend 3-dimensional orthogonal grid drawing of maximum degree 5 graphs. In addition 2-bend 3-dimensional grid drawings of the 6-regular multi-partite graphs are given.

A three-dimensional grid drawing of a graph G is a placement of the vertices at distinct integer points so that the straight-line segments representing the edges of G are pairwise non-crossing. It is shown that for any xed r 2, every r-colorable graph of n vertices has a three-dimensional grid drawing that ts into a box of volume O(n2). The order of magnitude of this bound cannot be improved.