Graph Treewidth and Geometric Thickness Parameters

Consider a drawing of a graph $G$ in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of $G$, is the classical graph parameter "thickness". By restricting the edges to be straight, we obtain the "geometric thickness". By further restricting the vertices to be in convex position, we obtain the "book thickness". This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth $k$, the maximum thickness and the maximum geometric thickness both equal $\lceil{k/2}\rceil$. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth $k$, the maximum book thickness equals $k$ if $k \leq 2$ and equals $k+1$ if $k \geq 3$. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity.Comment: A preliminary version of this paper appeared in the "Proceedings of the 13th International Symposium on Graph Drawing" (GD '05), Lecture Notes in Computer Science 3843:129-140, Springer, 2006. The full version was published in Discrete & Computational Geometry 37(4):641-670, 2007. That version contained a false conjecture, which is corrected on page 26 of this versio

Pursuit-Evasion in Graphs: Zombies, Lazy Zombies and a Survivor

We study zombies and survivor, a variant of the game of cops and robber on graphs. In this variant, the single survivor plays the role of the robber and attempts to escape from the zombies that play the role of the cops. The zombies are restricted, on their turn, to always follow an edge of a shortest path towards the survivor. Let z(G) be the smallest number of zombies required to catch the survivor on a graph G with n vertices. We show that there exist outerplanar graphs and visibility graphs of simple polygons such that z(G) = ?(n). We also show that there exist maximum-degree-3 outerplanar graphs such that z(G) = ?(n/log(n)). Let z_L(G) be the smallest number of lazy zombies (zombies that can stay still on their turn) required to catch the survivor on a graph G. We show that lazy zombies are more powerful than normal zombies but less powerful than cops. We prove that z_L(G) ? 2 for connected outerplanar graphs and this bound is tight in the worst case. We show that z_L(G) ? k for connected graphs with treedepth k. This result implies that z_L(G) is at most (k+1)log n for connected graphs with treewidth k, O(?n) for connected planar graphs, O(?{gn}) for connected graphs with genus g and O(h?{hn}) for connected graphs with any excluded h-vertex minor. Our results on lazy zombies still hold when an adversary chooses the initial positions of the zombies

Some Results on Minors for Graphs and Matroids.

This dissertation solves some problems relating to the theory of graphs. The first type of problem considered concerns the structure of various classes of graphs which arise naturally from outerplanar graphs. These problems are motivated by Chartrand and Harary\u27s well-known characterization of outerplanar graphs. This theorem states that K\sb4 and K\sb{2,3} are the only non-outerplanar graphs for which both G$\\$e, the deletion of the edge e from the graph G, and G/e, the contraction of the edge e, are outerplanar for all edges e of G. Following Gubser\u27s characterization of almost-planar graphs, we begin our study of graphs related to outerplanar graphs by characterizing the non-outerplanar graphs for which G$\\$e or G/e is outerplanar. We call these graphs almost-outerplanar (or 1-outerplanar). We then consider the corresponding problem for almost-outerplanar graphs and characterize the graphs G that are not almost-outerplanar such that G$\\$e or G/e is almost-outerplanar or outerplanar for every edge e of G. We end our study of graphs arising from outerplanar graphs by relaxing Chartrand and Harary\u27s condition characterizing outerplanarity in a different way. This time we describe the non-outerplanar graphs G for which G$\\$e and G/e are outerplanar for at least one edge e. The second problem we solve is motivated by Hartvigsen and Zemel\u27s characterization of graphs having the property that every circuit basis is fundamental. This theorem states that a graph has every circuit basis fundamental if and only if the graph has no minor isomorphic to one of five graphs. We consider the corresponding problem for binary matroids. We show that, in general, the class of binary matroids for which every circuit basis is fundamental is not closed under the taking of minors. However, this class is closed under the taking of series-minors. We also described some general properties of this class of matroids. We end this chapter by extending Hartvigsen and Zemel\u27s result to the class of regular matroids

Chromatic numbers of exact distance graphs

For any graph G = (V;E) and positive integer p, the exact distance-p graph G[\p] is the graph with vertex set V , which has an edge between vertices x and y if and only if x and y have distance p in G. For odd p, Nešetřil and Ossona de Mendez proved that for any fixed graph class with bounded expansion, the chromatic number of G[\p] is bounded by an absolute constant. Using the notion of generalised colouring numbers, we give a much simpler proof for the result of Nešetřil and Ossona de Mendez, which at the same time gives significantly better bounds. In particular, we show that for any graph G and odd positive integer p, the chromatic number of G[\p] is bounded by the weak (2