We investigate embeddings of graphs into the infinite extended grid graph. The problem was motivated by computation on adiabatic quantum computers, but it is related to a number of other well studied grid embedding problems. Such problems typically deal with representing vertices by grid points, and edges by grid paths, while minimizing some objective function such as the area or the maximum le...

An orthogonal drawing of a graph is an embedding of the graph in the two-dimensional grid such that edges are routed along grid-lines. In this paper we explore lower bounds for orthogonal graph drawings. We prove lower bounds on the number of bends and, when crossings are not allowed, also lower bounds on the size of the grid. Communicated by D. Wagner: submitted July 1997; revised November 199...

A bent Hamilton cycle in a grid graph is one in which each edge in a successive pair of edges lies in a different dimension. We show that the d-dimensional grid graph has a bent Hamilton cycle if some dimension is even and d ≥ 3, and does not have a bent Hamilton cycle if all dimensions are odd. In the latter case, we determine the conditions for when a bent Hamilton path exists. For the d-dime...

For given a pair of nodes in a graph, the minimum non-separating path problem looks for a minimum weight path between the two nodes such that the remaining graph after removing the path is still connected. The balanced connected bipartition (BCP2) problem looks for a way to bipartition a graph into two connected subgraphs with their weights as equal as possible. In this paper we present an algo...

F. Ruskey Joe Sawada y May 15, 2002 Abstra t A bent Hamilton y le in a grid graph is one in whi h ea h edge in a su essive pair of edges lies in a di erent dimension. We show that the d-dimensional grid graph has a bent Hamilton y le if some dimension is even and d 3, and does not have a bent Hamilton y le if all dimensions are odd. In the latter ase, we determine the onditions for when a bent ...

The directed graph reachability problem takes as input an n-vertex directed graph G = (V,E), and two distinguished vertices s and t. The problem is to determine whether there exists a path from s to t in G. This is a canonical complete problem for class NL. Asano et al. proposed an Õ( √ n) space and polynomial time algorithm for the directed grid and planar graph reachability problem. The main ...

--------------------------------------------------------------Abstract-----------------------------------------------------------The replica algorithms are built based on the topology of the data grid. The real representation of grid is general graph. So we propose an algorithm called Replica Placement in Graph Topology Grid (RPGTG) to place replicas on graph based data grids. In this paper, we...

A graph G = (V,E) is word-representable if there exists a word w over the alphabet V such that letters x and y, x 6= y, alternate in w if and only if (x, y) ∈ E. Halldórsson, Kitaev and Pyatkin have shown that a graph is word-representable if and only if it admits a so-called semi-transitive orientation. A corollary of this result is that any 3-colorable graph is wordrepresentable.Masárová Akro...

Given a graph G, the (graph theory) general position problem is to find the maximum number of vertices such that no three vertices lie on a common geodesic. This graph invariant is called the general position number (gpnumber for short) of G and denoted by gp(G). In this paper, the gp-number is determined for a large class of subgraphs of the infinite grid graph and for the infinite diagonal gr...

Given a plane graph G, we wish to nd a drawing of G in the plane such that the vertices of G are represented as grid points, and the edges are represented as straight-line segments between their endpoints without any edge-intersection. Such drawings are called planar straight-line drawings of G. An additional objective is to minimize the area of the rectangular grid in which G is drawn. In this...