A three-dimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line-segments representing the edges are pairwise non-crossing. A O(n volume bound is proved for three-dimensional grid drawings of graphs with bounded degree, graphs with bounded genus, and graphs with no bounded complete graph as a minor. The previous best bound for these graph families was O(n ). These results (partially) solve open problems due to Pach, Thiele, and Toth (1997) and Felsner, Liotta, and Wismath (2001).

In this paper we show that every 2-connected embedded planar graph with faces of sizes d!..... df has a simple cycle separator of size 1.58~,/dl 2 +...+d} and we give an almost linear time algorithm for finding these separators, O(m~(n. n)). We show that the new upper bound expressed as a function of IG { = ~/d ~ +..- + d} is no larger, up to a constant factor than previous bounds that where expressed in terms of ~S. v where d is the maximum face size and v is the number of vertices and is much smaller for many graphs. The algorithms developed are simpler than earlier algorithms in that they work directly with the planar graph and its dual. They need not construct or work with the face-incidence graph as in [Mi186, GM87, GM]. 1

Methods for solving sparse linear systems of equations can be categorized under two broad classes- direct and iterative. Direct methods are methods based on gaussian elimination. This report discusses one such direct method namely Nested dissection. Nested Dissection, originally proposed by Alan George, is a technique for solving sparse linear systems efficiently. This report is a survey of some of the work in the area of nested dissection and attempts to put it together using a common framework.

Pach and Tóth [15] proved that any n-vertex graph of genus g and maximum degree d has a planar crossing number at most c g dn, for a constant c>1. We improve on this results by decreasing the bound to O(dgn), if g = o(n), and to O(g²), otherwise, and also prove that our result is tight within a constant factor.

The girth of a graph G has been de ned as the length of a shortest cycle of G. We design an O(n 5=4 log n) algorithm for finding the girth of an undirected n-vertex planar graph, giving the first o(n 2 ) algorithm for this problem. Our approach combines several techniques such as graph separation, hammock decomposition, covering of a planar graph with graphs of small tree-width, and dynamic shortest path computation. We discuss extensions and generalizations of our result.

We prove separator theorems in which the size of the separator is minimized with respect to non-negative vertex costs. We show that for any planar graph G there exists a vertex separator of total vertex cost at most c qP v2V (G) (cost(v)) 2 and that this bound is optimal within a constant factor. Moreover such a separator can be found in linear time. This theorem implies a variety of other separation results discussed in the paper. We describe application of our separator theorems to graph embedding problems, graph pebbling, and multi-- commodity flow problems. 1 Introduction Background. A separator is a small set of vertices or edges whose removal divides a graph into two roughly equal parts. The existence of small separators for some important classes of graphs such as planar graphs can be used in the design of efficient divide-and-conquer algorithms for problems on such graphs. Formally, a separator theorem for a given class of graphs S states that any n-vertex graph from S ca...

A track layout of a graph consists of a vertex colouring, an edge colouring, and a total order of each vertex colour class such that between each pair of vertex colour classes, there is no monochromatic pair of crossing edges. This paper studies track layouts and their applications to other models of graph layout. In particular, we improve on the results of Enomoto and Miyauchi [SIAM J. Discrete Math., 1999] regarding stack layouts of subdivisions, and give analogous results for queue layouts. We solve open problems due to Felsner, Wismath, and Liotta [Proc. Graph Drawing, 2001] and Pach, Thiele, and Toth [Proc. Graph Drawing, 1997] concerning three-dimensional straight-line grid drawings. We initiate the study of three-dimensional polyline grid drawings and establish volume bounds within a logarithmic factor of optimal.