Elsevier

Discrete Applied Mathematics

Volume 281, 15 July 2020, Pages 162-171
Discrete Applied Mathematics

Complexity-separating graph classes for vertex, edge and total colouring

https://doi.org/10.1016/j.dam.2019.02.039Get rights and content

Abstract

Given a class A of graphs and a decision problem π belonging to NP, we say that a full complexity dichotomy of A was obtained if one describes a partition of A into subclasses such that π is classified as polynomial or NP-complete when restricted to each subclass. The concept of full complexity dichotomy is particularly interesting for the investigation of NP-complete problems: as we partition a class A into NP-complete subclasses and polynomial subclasses, it becomes clearer why the problem is NP-complete in A. The class C of graphs that do not contain a cycle with a unique chord was studied by Trotignon and Vušković who proved a structure theorem which led to solving the vertex-colouring problem in polynomial time. In the present survey, we apply the structure theorem to study the complexity of edge-colouring and total-colouring, and show that even for graph classes with strong structure and powerful decompositions, the edge-colouring problem may be difficult. We discuss several surprising complexity dichotomies found in subclasses of C, and the concepts of separating problem proposed by David S. Johnson and the dual concept of separating class.

Section snippets

NP-completeness ongoing guide

In the sixteenth edition of his NP-Completeness Column: An Ongoing Guide [30], David S. Johnson focused on graph restrictions and their effect, with emphasis on the restrictions to graph classes and how they affect the complexity of various NP-hard problems. Graph classes were selected because of their broad algorithmic significance. The presentation consisted of a summary table with 30 rows containing the selected classes of graphs, and 11 columns the first devoted to the complexity of

Full dichotomies

Given a class A of graphs and a graph (decision) problem π belonging to NP, we say that a full complexity dichotomy of A is obtained if one describes a partition of A into subclasses A1,A2, such that π is classified as polynomial or NP-complete when restricted to each subclass Ai. The concept of full complexity dichotomy is particularly interesting for the investigation of NP-complete problems: as we partition a class A into NP-complete subclasses and polynomial subclasses, it becomes clearer

Beyond vertex, edge, and total colourings

Unichord-free graphs proved to have a rich structure that can be used to obtain interesting results with respect to the study of the complexity of colouring problems. In the context of clique-colouring and biclique-colouring problems, a clique of a graph G is a maximal set of vertices that induces a complete subgraph of G with at least one edge. A biclique of G is a maximal set of vertices that induces a complete bipartite subgraph of G with at least one edge. A clique-colouring of G is a

Every graph is easy or hard

In 2014, Dániel Marx gave a plenary talk at the 9th International Colloquium on Graph Theory and Combinatorics, entitled Every Graph is Easy or Hard: Dichotomy Theorems for Graph Problems. He highlighted three features of dichotomy theorems: dichotomy theorems give good research programmes, easy to formulate, but can be hard to complete; the search for dichotomy theorems may uncover algorithmic results that no one has thought of; proving dichotomy theorems may require good command of both

Acknowledgements

I am grateful to Guillermo Durán, Mario Valencia-Pabon and the Programme Chairs of LAGOS 2017, for the opportunity of giving this invited talk. Special thanks are due to my co-authors: I have benefited enormously from our collaborations; I have freely adapted material from our joint work. Alexsander A. de Melo gave me invaluable help to update Table 3.

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  • Cited by (0)

    This paper is based on an invited talk given at LAGOS 2017, the Latin-American Algorithms, Graphs and Optimization Symposium. This work is partially supported by Brazilian agencies CNPq and FAPERJ .

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