Refining the Hierarchies of Classes of Geometric Intersection Graphs

Abstract

We analyse properties of geometric intersection graphs to show strict containment between some natural classes of geometric intersection graphs. In particular, we show the following properties:• A graph G is outerplanar if and only if the 1-subdivision of G is outer-segment.• For each integer k ≥ 1, the class of intersection graphs of segments with k different lengths is a strict subclass of the class of intersection graphs of segments with k + 1 different lengths.• For each integer k ≥ 1, the class of intersection graphs of disks with k different sizes is a strict subclass of the class of intersection graphs of disks with k + 1 different sizes.• The class of outer-segment graphs is a strict subclass of the class of outer-string graphs.

Stick Graphs with Length Constraints

“…For a given collection S of geometric objects, the intersection graph of S has S as its vertex set and an edge whenever S ∩ S = ∅, for S, S ∈ S. This paper concerns recognition problems for classes of intersection graphs of restricted geometric objects, i.e., determining whether a given graph is an intersection graph of a family of restricted sets of geometric objects. A classic (general) class of intersection graphs is that of segment graphs, the intersection graphs of line segments in the plane 1 . For example, segment graphs are known to include planar graphs [4].…”

“…acknowledge support from DFG grants WO 758/11-1 and WO 758/9-1. 1 We follow the common convention that parallel segments do not intersect and each point in the plane belongs to at most two segments. 2 Note that ∃R includes NP, see [22,24] for background on the complexity class ∃R.…”

“…The simplicity of bipartite permutation graphs leads to a simpler linear-time recognition algorithm [27] than that of permutation graphs. Several recent articles [1,2,6,7] compare and study the geometric intersection graph classes occurring between the simple classes, such as bipartite permutation graphs, and the general classes, such a segment graphs. Cabello and Jejčič [1] mention that studying such classes with constraints on the sizes or lengths of the objects is an interesting direction for future work (and such constraints are the focus of our work).…”

“…Several recent articles [1,2,6,7] compare and study the geometric intersection graph classes occurring between the simple classes, such as bipartite permutation graphs, and the general classes, such a segment graphs. Cabello and Jejčič [1] mention that studying such classes with constraints on the sizes or lengths of the objects is an interesting direction for future work (and such constraints are the focus of our work). Note that similar length restrictions have been considered for other geometric intersection graphs such as interval graphs [15,16,23].…”

“…We provide the first efficient algorithm for STICK A and a faster algorithm for STICK AB ; see Section 2. Then we investigate the direction suggested by Cabello and Jejčič [1] where specific lengths are given for the segments of each vertex. In particular, this can be thought of as generalizing from unit stick graphs (i.e., bipartite permutation graphs), where every segment has the same length.…”