
Capturing Polynomial Time using Modular Decomposition
The question of whether there is a logic that captures polynomial time i...
read it

The use of a pruned modular decomposition for Maximum Matching algorithms on some graph classes
We address the following general question: given a graph class C on whic...
read it

(k)critical trees and kminimal trees
In a graph G=(V,E), a module is a vertex subset M of V such that every v...
read it

Modular decomposition of transitive graphs and transitively orienting their complements
The modular decomposition of a graph is a canonical representation of it...
read it

On rectangledecomposable 2parameter persistence modules
This paper addresses two questions: (1) can we identify a sensible class...
read it

Regular cylindrical algebraic decomposition
We show that a strong wellbased cylindrical algebraic decomposition P o...
read it

TriPartitions and Bases of an Ordered Complex
Generalizing the decomposition of a connected planar graph into a tree a...
read it
(α, β)Modules in Graphs
Modular Decomposition focuses on repeatedly identifying a module M (a collection of vertices that shares exactly the same neighbourhood outside of M) and collapsing it into a single vertex. This notion of exactitude of neighbourhood is very strict, especially when dealing with real world graphs. We study new ways to relax this exactitude condition. However, generalizing modular decomposition is far from obvious. Most of the previous proposals lose algebraic properties of modules and thus most of the nice algorithmic consequences. We introduce the notion of an (α, β)module, a relaxation that allows a bounded number of errors in each node and maintains some of the algebraic structure. It leads to a new combinatorial decomposition with interesting properties. Among the main results in this work, we show that minimal (α, β)modules can be computed in polynomial time, and that every graph admits an (α,β)modular decomposition tree, thus generalizing Gallai's Theorem (which corresponds to the case for α = β = 0). Unfortunately we give evidence that computing such a decomposition tree can be difficult.
READ FULL TEXT
Comments
There are no comments yet.