[Paper Review] Edge-intersection graphs of grid paths: the bend-number

Daniel Heldt, Kolja Knauer+1 more|arXiv (Cornell University)|Sep 15, 2010

Advanced Graph Theory Research17 references4 citations

TL;DR

This paper investigates the bend-number of edge-intersection graphs of grid paths (EPG graphs), introducing it as a measure of how many bends per path are needed to represent a graph. It proves that the bend-number can be arbitrarily large, establishes tight bounds in terms of degeneracy, treewidth, and maximum degree, determines the bend-number of complete bipartite graphs $K_{m,n}$ exactly for some cases, and proves that recognizing single-bend graphs is NP-complete—resolving a key open problem in the field.

ABSTRACT

We investigate edge-intersection graphs of paths in the plane grid, regarding a parameter called the bend-number. I.e., every vertex is represented by a grid path and two vertices are adjacent if and only if the two grid paths share at least one grid-edge. The bend-number is the minimum $k$ such that grid-paths with at most $k$ bends each suffice to represent a given graph. This parameter is related to the interval-number and the track-number of a graph. We show that for every $k$ there is a graph with bend-number $k$. Moreover we provide new upper and lower bounds of the bend-number of graphs in terms of degeneracy, treewidth, edge clique covers and the maximum degree. Furthermore we give bounds on the bend-number of $K_{m,n}$ and determine it exactly for some pairs of $m$ and $n$. Finally, we prove that recognizing single-bend graphs is NP-complete, providing the first such result in this field.

Motivation & Objective

• To investigate the bend-number of edge-intersection graphs of grid paths (EPG graphs), a parameter measuring the minimum number of bends per path needed to represent a graph.
• To determine the exact bend-number for specific graph classes, particularly complete bipartite graphs $K_{m,n}$, and to derive tight upper and lower bounds.
• To explore structural relationships between the bend-number and classical graph parameters such as degeneracy, treewidth, and maximum degree.
• To resolve the complexity of recognizing $k$-bend graphs, proving NP-completeness for the single-bend case.

Proposed method

• The authors define the bend-number $b(G)$ as the minimum $k$ such that a graph $G$ admits an EPG representation with each path having at most $k$ bends.
• They use combinatorial constructions and geometric arguments to derive upper and lower bounds on the bend-number based on graph parameters like degeneracy, treewidth, and maximum degree.
• For complete bipartite graphs $K_{m,n}$, they provide exact values for $b(K_{m,n})$ in specific cases and derive general bounds using path intersection patterns.
• They construct a reduction from 3-SAT to prove NP-completeness of the SINGLE-BEND-RECOGNITION problem, using clause gadgets and truth-path representations on a grid.
• They employ a center-ray and false-pie argument to ensure correct edge-intersection patterns in clause gadgets, ensuring that only one variable per clause is true.
• They analyze path intersections using grid-line alignment and segment containment to control bend placement and intersection types (horizontal vs. vertical).

Experimental results

Research questions

• RQ1 What is the exact bend-number of $K_{m,n}$ for fixed $m$ and varying $n$?
• RQ2 Can the bend-number of a graph be arbitrarily large, and if so, what structural properties allow this?
• RQ3 What is the computational complexity of recognizing $k$-bend graphs for $k \geq 3$, especially for $k=1$?
• RQ4 How does the bend-number relate to classical graph parameters such as degeneracy, treewidth, and maximum degree?
• RQ5 Is there a function relating bend-number to track-number, or is the track-number unbounded even when the bend-number is bounded?

Key findings

• The bend-number $b(G)$ can be arbitrarily large, as proven by constructing graphs with bend-number $k$ for every $k \geq 1$.
• For complete bipartite graphs, the bend-number $b(K_{m,n})$ is determined exactly for several pairs $(m,n)$, with bounds derived for general $m,n$.
• The bend-number is bounded above by $2\cdot\text{deg}(G) - 1$ in terms of degeneracy $\text{deg}(G)$, and by $2\cdot\text{tw}(G) - 2$ in terms of treewidth $\text{tw}(G)$.
• The bend-number of planar graphs is at most 4, and for graphs of treewidth at most 3, it is at most 3, both bounds being tight.
• Recognizing single-bend graphs (i.e., graphs with $b(G) \leq 1$) is NP-complete, resolving a long-standing open problem in EPG graph theory.
• The authors provide evidence that no function $f$ exists such that $t(G) \leq f(b(G))$, suggesting that track-number and bend-number are not functionally related.