In a convex grid drawing of a plane graph, every edge is drawn as a straight-line segment without any edge-intersection, every vertex is located at a grid point, and every facial cycle is drawn as a convex polygon. A plane graph G has a convex drawing if and only if G is internally triconnected. It has been known that an internally triconnected plane graph G of n vertices has a convex grid draw...

Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an l × l grid minor is exponential in l. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A grid-like-minor of order l in a graph G is a set of paths in Gwhose intersection graph is bipart...

Fix g > 1. Every graph of large enough tree-width contains a g × g grid as a minor; but here we prove that every four-edge-connected graph of large enough tree-width contains a g × g grid as an immersion (and hence contains any fixed graph with maximum degree at most four as an immersion). This result has a number of applications.

We use basic results from graph theory to design two algorithms for constructing 3-dimensional, intersection-free orthogonal grid drawings of n vertex graphs of maximum degree 6. Our first algorithm gives drawings bounded by an O(x/~ ) • O(x/~ • O(v~ ) box; each edge route contains at most 7 bends. The best previous result generated edge routes containing up to 16 bends per route. Our second al...

A touching triangle graph (TTG) representation of a planar graph is a planar drawing Γ of the graph, where each vertex is represented as a triangle and each edge e is represented as a side contact of the triangles that correspond to the end vertices of e. We call Γ a proper TTG representation if Γ determines a tiling of a triangle, where each tile corresponds to a distinct vertex of the input g...

A touching triangle graph representation (TTG) of a planar graph is a planar drawing Γ of the graph, where each vertex is represented as a triangle and each edge e is represented as a side contact of the triangles that correspond to the endvertices of e. We call Γ a proper TTG if Γ determines a tiling of a triangle, where each tile corresponds to a distinct vertex of the input graph. In this pa...

Monotone grid classes of permutations have proven very effective in helping to determine structural and enumerative properties of classical permutation pattern classes. Associated with grid class Grid(M) is a graph, G(M), known as its “row-column” graph. We prove that the exponential growth rate of Grid(M) is equal to the square of the spectral radius of G(M). Consequently, we utilize spectral ...