Arrangement Graphs and Intersection Graphs of Curves

Abstract

An arrangement of a set of non-self-intersecting curves is their embedding in the Euclidean plane such that at each intersection point the curves involved cross each other. Arrangements are well-studied both in Discrete Geometry and Computational Geometry. They serve as natural sources of graphs. In this thesis, we investigate two such graph classes that naturally arise from arrangements of curves.

The first graph class, called arrangement graphs, arises directly from the embedding itself by considering the intersection points as vertices and the segments of curves whose endpoints are the only vertices in it are the edges of the graph. The second graph class, called intersection graphs of finite curves, is obtained by treating each curve as a vertex and two vertices are adjacent if their corresponding curves intersect.

We begin with arrangement graphs. A pseudoline is a curve that extends to infinity in both ends. A pseudoline arrangement is a collection of at least three pairwise crossing pseudolines. It is simple if no three pseudolines meet at a point. A pseudoline arrangement graph is obtained from a simple pseudoline arrangement. First, we solve the Pseudoline Arrangement Graph Realization Problem, that is, given a sequence of finite numbers whether there is a pseudoline arrangement graph whose list of degrees of its vertices matches with the input sequence. And if the answer is yes, then we construct the pseudoline arrangement graph. Second, we study the eccentricities of vertices in pseudoline arrangement graphs. In particular, we find the diameter of such graphs, and then characterize the vertices that have eccentricity as the diameter.

Next, we move on to intersection graphs. A string graph is the intersection graph of finite curves. When these finite curves pairwise intersect at most once, we get 1-string graphs. An outerstring graph is a string graph obtained from the arrangement of finite curves in a disk such that an endpoint of each curve lies on the boundary of the disk. Kostochka and Nesetril studied coloring 1-string graphs with girth five to address questions of Erdos and of Kratochvil and Nesetril. In an attempt to improve these bounds we prove that outerstring graphs with girth g >= 5 and a minimum degree of at least 2 have a chain of (g-4) vertices with degree 2. This generalizes results of Ageev and of Esperet and Ochem on circle graphs: intersection graphs of chords of a circle. Our result also implies that outerstring graphs with a girth of at least five are 3-colorable, improving the previous bound of five by Gavenciak et al.

Next, we consider a geometric analog of string graphs called bend graphs (popularly known as VPG): intersection graphs of rectilinear curves. Upon fixing these rectilinear curves to be alternating horizontal and vertical line segments in the plane of at most k + 1 line segments, we obtain the graph class B(k). Clearly, B(k) are also string graphs. However, every string graph also belongs to B(k), for some k. The smallest value of k such that a string graph G \in B(k) is called its bend number. This parameter gives us a way of categorizing string graphs. Obviously B(k) \subseteq B(k+1). Chaplick et al. were the first to provide such a separating example proving that B(k) \notsubseteq B(k+1) for all k \in N, and further asked for chordal separating examples. In an attempt to answer their question, we show that there are infinitely many values of k such that there are separating examples for B(k) \notsubseteq B(k+1) that are split graphs (which are also chordal). We also prove similar results for a different variant of bend number called the proper bend number.

Finally, we close by highlighting the use of techniques developed for arrangement graphs that help to solve problems of string graphs. In particular, we reprove the result of Kostochka and Nesetril on coloring 1-string graphs with girth five by using the techniques developed while solving the pseudoline arrangement graph realization problem.