Joan P. Hutchinson

A Separator Theorem for Graphs of Bounded Genus

Many divide-and-conquer algorithms on graphs are based on finding a small set of vertices or edges whose removal divides the graph roughly in half. Most graphs do not have the necessary small separators, but some useful classes do. One such class is planar graphs: If we can draw an n-vertex graph on the plane, then we can bisect it by removing

On representations of some thickness-two graphs

O(\sqrt{n})

Rectangle-visibility representations of bipartite graphs

vertices [Lipt79b]. The main result of this paper is that if we can draw a graph on a surface of genus g, then we can bisect it by removing

On Rectangle Visibility Graphs

O(\sqrt{gn})

Three-Coloring Graphs Embedded on Surfaces with All Faces Even-Sided

vertices. This bound is best possible to within a constant factor. We give an algorithm for finding the separator that takes time linear in the number of edges in the graph, given an embedding of the graph in its genus surface. We discuss some extensions and applications of these results.

On the thickness and arboricity of a graph

This paper considers representations of graphs as rectanglevisibility graphs and as doubly linear graphs. These are, respectively, graphs whose vertices are isothetic rectangles in the plane with adjacency determined by horizontal and vertical visibility, and graphs that can be drawn as the union of two straight-edged planar graphs. We prove that these graphs have, with n vertices, at most 6n−20 (resp., 6n−18) edges, and we provide examples of these graphs with 6n−20 edges for each n≥8.

Colouring Eulerian Triangulations

Abstract The paper considers representations of bipartite graphs as rectangle-visibility graphs , i.e., graphs whose vertices are rectangles in the plane, with adjacency determined by horizontal and vertical visibility. It is shown that, for p ⩽ q , K p , q has a representation with no rectangles having collinear sides if and only if p ⩽ 2 or p = 3 and q ⩽ 4. More generally, it is shown that K p , q is a rectangle-visibility graph if and only if p ⩽ 4. Finally, it is shown that every bipartite rectangle-visibility graph on n ⩾ 4 vertices has at most 4 n − 12 edges.

On the independence ratio of a graph

We study the problem of drawing a graph in the plane so that the vertices of the graph are rectangles that are aligned with the axes, and the edges of the graph are horizontal or vertical lines-of-sight. Such a drawing is useful, for example, when the vertices of the graph contain information that we wish displayed on the drawing; it is natural to write this information inside the rectangle corresponding to the vertex. We call a graph that can be drawn in this fashion a rectangle-visibility graph, or RVG. Our goal is to find classes of graphs that are RVGs. We obtain several results: 1. For 1 ≤ k ≤ 4, k-trees are RVGs. 2. Any graph that can be decomposed into two caterpillar forests is an RVG. 3. Any graph whose vertices of degree four or more form a distance-two independent set is an RVG. 4. Any graph with maximum degree four is an RVG. Our proofs are constructive and yield linear-time layout algorithms.

On eulerian circuits and words with prescribed adjacency patterns

Every graph embedded on a surface of positive genus with every face bounded by an even number of edges can be 3-colored provided all noncontractible cycles in the graph are sufficiently long. The bound of three colors is the smallest possible for this type of result.

Fractional chromatic numbers of cones over graphs

We prove that the thickness and the arboricity of a graph with e edges are at most ⌊e3 + 32⌋ and ⌈e2⌉, respectively, and that the latter bound is best possible.