Let G be a Q4-free median graph on n vertices and m edges. Let k be the number of equivalence classes of Djoković-Winkler’s relation Θ and let h be the number of Q3’s in G. Then we prove that 2n −m − k + h = 2. We also characterize median grid graphs in several different ways, for instance, they are the grid graphs with m− n + 1 squares. To obtain these results we introduce the notion of square...

In this paper we present a generalization of the shift method algorithm [4, 6] to obtain a straight-line grid drawing of a triconnected graph, where vertex representations have a certain speci ed size. We propose vertex representations having a rectangular shape. Additionally, one may demand maintainance of the criterion of strong visibility, that is, any possible line segment connecting two ad...

Though our main technical result concerns random graphs in the G(n,p) model, let us mention other contexts in which threshold phenomena occur. One classical example is Percolation, an area started in physics. A typical question here is this: given a planar grid and 0 < p < 1. Create a graph by keeping each edge of the planar grid with probability p and removing each edge with probability 1-p. T...

We consider an optimization version of the image segmentation problem, in which we are given a grid graph with weights on the grid cells. We are interested in finding the maximum weight subgraph such that the subgraph can be decomposed into two ”star-shaped” images. We show that this problem can be reduced to the problem of finding a maximum-weight closed set in an appropriately defined directe...

For some search problems, the graph is known beforehand and there is time to preprocess the graph to make the search faster. One such example is video games, where one can often preprocess maps before a game is released or while a map is loaded into memory. The data produced by preprocessing should use only a small amount of memory, and, in case they are generated during runtime, preprocessing ...

In this paper, we prove that every planar 4-connected graph has a CZ-representation—a string representation using paths in a rectangular grid that contain at most one vertical segment. Furthermore, two paths representing vertices u, v intersect precisely once whenever there is an edge between u and v. The required size of the grid is n× 2n.

An odd neighborhood transversal of a graph is a set of its vertices that intersects the set of neighbors of each of its vertices in an odd number of elements. In the case of grid graphs this odd number will be either one or three. We characterize those grid graphs that have odd neighborhood transversals.

We define the visual complexity of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw two collinear edges of the same vertex). Let n denote the number of vertices of a graph. We show that trees can be drawn with 3n/4 straight-line segments on a ...

In this paper, we propose a new global routing methodology named SEGRoute, which is based on grid graph segmentation of the given routing problem. The SEGRouting methodology first divides the grid graph of the routing problem into various segments; it then performs routing for the various segments (completing routing if possible and trying to maintain minimum wirelength) and then performs a “se...

In this paper we study a trail routing and a hamiltonian cycle in a class of grid graphs, polycube and polymino. A Spanning closed trail is an eulerian subgraph containing all vertices of a given graph. For general grid graphs we prove that the problem of nding that trail is NP-complete and for a wide subclass of grid graphs, called polymino, we give an optimal algorithm if it exists. For polyc...