The \emph{segment number} of a planar graph is the smallest number of line segments whose union represents a crossing-free straight-line drawing of the given graph in the plane. The segment number is a measure for the visual complexity of a drawing; it has been studied extensively.
In this paper, we study three variants of the segment number: for planar graphs, we consider crossing-free polyline drawings in 2D; for arbitrary graphs, we consider crossing-free straight-line drawings in 3D and straight-line drawings with crossings in 2D. We first construct an infinite family of planar graphs where the classical segment number is asymptotically twice as large as each of the new variants of the segment number. Then we establish the $\exists\mathbb{R}$-completeness (which implies the NP-hardness) of all variants. Finally, for cubic graphs, we prove lower and upper bounds on the new variants of the segment number, depending on the connectivity of the given graph.
A visibility representation is a classical drawing style of planar graphs. It displays the vertices of a graph as horizontal vertex-segments, and each edge is represented by a vertical edge-segment touching the segments of its end vertices; beyond that segments do not intersect. We generalize visibility to 1-visibility, where each edge- (vertex-) segment crosses at most one vertex- (edge-) segment. In other words, a vertex is crossed by at most one edge, and vice-versa. We show that 1-visibility properly extends 1-planarity and develop a linear time algorithm to compute a 1-visibility represen…
In this survey paper, we present open problems and conjectures on visibility graphs of points, segments and polygons along with necessary backgrounds for understanding them.
We prove that for any triangle-free intersection graph of $n$ axis-parallel segments in the plane, the independence number $α$ of this graph is at least $α\ge n/4 + Ω(\sqrt{n})$. We complement this with a construction of a graph in this class satisfying $α\le n/4 + c \sqrt{n}$ for an absolute constant $c$, which demonstrates the optimality of our result.
We consider several classes of intersection graphs of line segments in the plane and prove new equality and separation results between those classes. In particular, we show that: (1) intersection graphs of grounded segments and intersection graphs of downward rays form the same graph class, (2) not every intersection graph of rays is an intersection graph of downward rays, and (3) not every intersection graph of rays is an outer segment graph. The first result answers an open problem posed by Cabello and Jejčič. The third result confirms a conjecture by Cabello. We thereby completely elucidate…
In the 1970s, Erdos asked whether the chromatic number of intersection graphs of line segments in the plane is bounded by a function of their clique number. We show the answer is no. Specifically, for each positive integer $k$, we construct a triangle-free family of line segments in the plane with chromatic number greater than $k$. Our construction disproves a conjecture of Scott that graphs excluding induced subdivisions of any fixed graph have chromatic number bounded by a function of their clique number.
A segment representation of a graph is an assignment of line segments in 2D to the vertices in such a way that two segments intersect if and only if the corresponding vertices are adjacent. Not all graphs have such segment representations, but they exist, for example, for all planar graphs. In this note, we study the resolution that can be achieved for segment representations, presuming the ends of segments must be on integer grid points. We show that any planar graph (and more generally, any graph that has a so-called $L$-representation) has a segment representation in a grid of width and hei…
The visual complexity of a graph drawing can be measured by the number of geometric objects used for the representation of its elements. In this paper, we study planar graph drawings where edges are represented by few segments. In such a drawing, one segment may represent multiple edges forming a path. Drawings of planar graphs with few segments were intensively studied in the past years. However, the area requirements were only considered for limited subclasses of planar graphs. In this paper, we show that trees have drawings with $3n/4-1$ segments and $n^2$ area, improving the previous resul…
We study the problem of ordered stabbing of $n$ balls (of arbitrary and possibly different radii, no ball contained in another) in $\mathbb{R}^d$, $d \geq 3$, with either a directed line segment or a (directed) polygonal curve. Here, the line segment, respectively polygonal curve, shall visit (intersect) the given sequence of balls in the order of the sequence. We present a deterministic algorithm that decides whether there exists a line segment stabbing the given sequence of balls in order, in time $O(n^{4d-2} \log n)$. Due to the descriptional complexity of the region containing these line s…
A disk graph is the intersection graph of disks in the plane, a unit disk graph is the intersection graph of same radius disks in the plane, and a segment graph is an intersection graph of line segments in the plane. It can be seen that every disk graph can be realized by disks with centers on the integer grid and with integer radii; and similarly every unit disk graph can be realized by disks with centers on the integer grid and equal (integer) radius; and every segment graph can be realized by segments whose endpoints lie on the integer grid. Here we show that there exist disk graphs on $n$ …
In this paper, we determine the computational complexity of recognizing two graph classes, \emph{grounded L}-graphs and \emph{stabbable grid intersection} graphs. An L-shape is made by joining the bottom end-point of a vertical ($\vert$) segment to the left end-point of a horizontal ($-$) segment. The top end-point of the vertical segment is known as the {\em anchor} of the L-shape. Grounded L-graphs are the intersection graphs of L-shapes such that all the L-shapes' anchors lie on the same horizontal line. We show that recognizing grounded L-graphs is NP-complete. This answers an open questio…
Planar graphs are known to allow subexponential algorithms running in time $2^{O(\sqrt n)}$ or $2^{O(\sqrt n \log n)}$ for most of the paradigmatic problems, while the brute-force time $2^{Θ(n)}$ is very likely to be asymptotically best on general graphs. Intrigued by an algorithm packing curves in $2^{O(n^{2/3}\log n)}$ by Fox and Pach [SODA'11], we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs) or segments (segment intersection graphs) and which problems have no such algorithms under the ETH (Exponential Time Hypothesis). Among …
The segment number of a planar graph $G$ is the smallest number of line segments needed for a planar straight-line drawing of $G$. Dujmović, Eppstein, Suderman, and Wood [CGTA'07] introduced this measure for the visual complexity of graphs. There are optimal algorithms for trees and worst-case optimal algorithms for outerplanar graphs, 2-trees, and planar 3-trees. It is known that every cubic triconnected planar $n$-vertex graph (except $K_4$) has segment number $n/2+3$, which is the only known universal lower bound for a meaningful class of planar graphs. We show that every triconnected plana…
We show that the following variant of labeling rotating maps is NP-hard, and present a polynomial approximation scheme for solving it. The input is a set of feature points on a map, to each of which a vertical bar of zero width is assigned. The goal is to choose the largest subsets of the bars such that when the map is rotated and the labels remain vertical, none of the bars intersect. We extend this algorithm to the general case where labels are arbitrary objects.
Let $P$ be a set of $n$ points in strictly convex position in the plane. Let $D_n$ be the graph whose vertex set is the set of all line segments with endpoints in $P$, where disjoint segments are adjacent. The chromatic number of this graph was first studied by Araujo, Dumitrescu, Hurtado, Noy, and Urrutia [2005] and then by Dujmović and Wood [2007]. Improving on their estimates, we prove the following exact formula: $$χ(D_n) = n - \left\lfloor \sqrt{2n + \tfrac{1}{4}} - \tfrac{1}{2}\right\rfloor.$$
We explore what could make recognition of particular intersection-defined classes hard. We focus mainly on unit grid intersection graphs (UGIGs), i.e., intersection graphs of unit-length axis-aligned segments and grid intersection graphs (GIGs, which are defined like UGIGs without unit-length restriction) and string graphs, intersection graphs of arc-connected curves in a plane. We show that the explored graph classes are NP-hard to recognized even when restricted on graphs with arbitrarily large girth, i.e., length of a shortest cycle. As well, we show that the recognition of these classes re…
A graph $G$ with vertex set $\{v_1,v_2,\ldots,v_n\}$ is an intersection graph of segments if there are segments $s_1,\ldots,s_n$ in the plane such that $s_i$ and $s_j$ have a common point if and only if $\{v_i,v_j\}$ is an edge of~$G$. In this expository paper, we consider the algorithmic problem of testing whether a given abstract graph is an intersection graph of segments. It turned out that this problem is complete for an interesting recently introduced class of computational problems, denoted by $\exists\mathbb{R}$. This class consists of problems that can be reduced, in polynomial time, t…
Let $P$ be a set of $n$ points in general and convex position in the plane. Let $D_n$ be the graph whose vertex set is the set of all line segments with endpoints in $P$, where disjoint segments are adjacent. The chromatic number of this graph was first studied by Araujo et al. [\emph{CGTA}, 2005]. The previous best bounds are $\frac{3n}{4}\leqχ(D_n) <n-\sqrt{\frac{n}{2}}$ (ignoring lower order terms). In this paper we improve the lower bound to $χ(D_n)\geq n-\sqrt{2n}$, to conclude a near-tight bound on $χ(D_n)$.
