# Orthogonal Tree Decompositions of Graphs

## TL;DR

This paper explores graphs with two tree decompositions having bounded intersections, establishing new bounds on treewidth and crossing numbers, and analyzing classes like minor-closed, string, and series parallel graphs.

## Contribution

It introduces the concept of orthogonal tree decompositions with bounded intersections and proves their existence for various graph classes, deriving bounds on treewidth and crossing numbers.

## Key findings

- Graphs with two such decompositions have $O(\sqrt{n})$ treewidth.
- Proper minor-closed classes have bounded intersection decompositions.
- Series parallel graphs do not necessarily have such decompositions.

## Abstract

This paper studies graphs that have two tree decompositions with the property that every bag from the first decomposition has a bounded-size intersection with every bag from the second decomposition. We show that every graph in each of the following classes has a tree decomposition and a linear-sized path decomposition with bounded intersections: (1) every proper minor-closed class, (2) string graphs with a linear number of crossings in a fixed surface, (3) graphs with linear crossing number in a fixed surface. Here `linear size' means that the total size of the bags in the path decomposition is $O(n)$ for $n$-vertex graphs. We then show that every $n$-vertex graph that has a tree decomposition and a linear-sized path decomposition with bounded intersections has $O(\sqrt{n})$ treewidth. As a corollary, we conclude a new lower bound on the crossing number of a graph in terms of its treewidth. Finally, we consider graph classes that have two path decompositions with bounded intersections. Trees and outerplanar graphs have this property. But for the next most simple class, series parallel graphs, we show that no such result holds.