Efficient Algorithms for Determining Elimination Distance to Bounded Degree on Planar Graphs

Core Concepts

The elimination distance of a graph to the class of bounded degree graphs can be efficiently computed on planar graphs.

Abstract

The paper studies the graph parameter "elimination distance to bounded degree", which was introduced by Bulian and Dawar in their work on the parameterized complexity of the graph isomorphism problem.

The main result is that the problem of determining the elimination distance of a planar graph to the class of graphs with maximum degree at most d is fixed-parameter tractable, with the parameters being the elimination distance and the degree bound d.

The key steps in the proof are:

1. If the input graph G has small treewidth, the property of having elimination distance at most k to the class of graphs with maximum degree at most d can be efficiently tested using Courcelle's Theorem.

2. If G has large treewidth, it must contain a large grid minor. The authors show that in this case, either G does not belong to the class Ck,d (if too many branch sets contain high-degree vertices), or there exists an "irrelevant" vertex whose deletion does not change the membership in Ck,d. By iteratively removing such irrelevant vertices, the algorithm either reaches a small treewidth instance or concludes that G ∉ Ck,d.

The algorithm runs in time f(k, d) · nc for a computable function f and constant c, improving to f(k, d) · n³ when the input graph is planar.

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Key Insights Distilled From

Elimination distance to bounded degree on planar graphs

by Alexander Li... at arxiv.org 04-04-2024

https://arxiv.org/pdf/2007.02413.pdf

Deeper Inquiries

How can the techniques developed in this paper be extended to handle more general graph classes beyond planar graphs

The techniques developed in the paper can be extended to handle more general graph classes beyond planar graphs by considering classes characterized by the exclusion of specific graph minors or topological minors. By adapting the algorithm to identify and work with minor models of these excluded graphs, the elimination distance problem can be efficiently solved on a broader range of graph classes. This extension would involve modifying the algorithm to account for the specific properties and structures of the excluded minors, ensuring that the elimination process is tailored to the characteristics of these graph classes.

Can the elimination distance problem be solved efficiently on graph classes characterized by the exclusion of a family of finite graphs as topological minors

Yes, the elimination distance problem can be efficiently solved on graph classes characterized by the exclusion of a family of finite graphs as topological minors. Recent research has shown that there exists a fixed-parameter tractable algorithm for computing elimination distance to classes characterized by the exclusion of a family of finite graphs as topological minors. By leveraging the properties of these excluded graphs and developing specialized techniques to handle the elimination process within this context, efficient solutions can be obtained for such graph classes. This advancement opens up new possibilities for addressing the elimination distance problem on a wider range of graph classes with diverse structural constraints.

What other applications of the elimination distance parameter can be explored, beyond the graph isomorphism problem studied by Bulian and Dawar

Beyond the graph isomorphism problem studied by Bulian and Dawar, the elimination distance parameter has various other applications in the field of graph theory and algorithm design. One potential application is in the development of efficient algorithms for solving optimization problems on graph classes with bounded elimination distance. By utilizing the elimination distance parameter as a measure of proximity to specific graph classes, algorithms can be tailored to exploit this proximity and achieve improved computational efficiency. Additionally, the elimination distance parameter can be used in the analysis of structural properties of graphs and in studying the complexity of graph problems in the context of distance to specific graph classes. This parameter provides a versatile tool for understanding the structural complexity of graphs and designing algorithms that leverage this structural information for optimization and decision-making tasks.