DeepAI

# Recognition and Drawing of Stick Graphs

A Stick graph is an intersection graph of axis-aligned segments such that the left end-points of the horizontal segments and the bottom end-points of the vertical segments lie on a `ground line,' a line with slope -1. It is an open question to decide in polynomial time whether a given bipartite graph G with bipartition A∪ B has a Stick representation where the vertices in A and B correspond to horizontal and vertical segments, respectively. We prove that G has a Stick representation if and only if there are orderings of A and B such that G's bipartite adjacency matrix with rows A and columns B excludes three small `forbidden' submatrices. This is similar to characterizations for other classes of bipartite intersection graphs. We present an algorithm to test whether given orderings of A and B permit a Stick representation respecting those orderings, and to find such a representation if it exists. The algorithm runs in time linear in the size of the adjacency matrix. For the case when only the ordering of A is given, we present an O(|A|^3|B|^3)-time algorithm. When neither ordering is given, we present some partial results about graphs that are, or are not, Stick representable.

• 9 publications
• 8 publications
• 45 publications
• 23 publications
• 30 publications
05/18/2022

### On the complexity of recognizing Stick graphs

Stick graphs are defined as follows. Let A (respectively B) be a set of ...
07/11/2019

### Stick Graphs with Length Constraints

Stick graphs are intersection graphs of horizontal and vertical line seg...
07/15/2020

### Plattenbauten: Touching Rectangles in Space

Planar bipartite graphs can be represented as touching graphs of horizon...
03/15/2021

### A fast and scalable bottom-left-fill algorithm to solve nesting problems using a semi-discrete representation

We present a fast algorithm to solve nesting problems based on a semi-di...
07/21/2020

### Stick graphs: examples and counter-examples

Stick graphs are the intersection graphs of vertical and horizontal segm...
09/05/2022

### Recognizing Geometric Intersection Graphs Stabbed by a Line

In this paper, we determine the computational complexity of recognizing ...
06/23/2021

### Forced pairs in A-Stick graphs

A Stick graph G=(A∪B, E) is the intersection graph of a set A of horizon...