Complexity-separating graph classes for vertex, edge and total colouring☆
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 of graphs and a graph (decision) problem belonging to NP, we say that a full complexity dichotomy of is obtained if one describes a partition of into subclasses such that 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 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 is a maximal set of vertices that induces a complete subgraph of with at least one edge. A biclique of is a maximal set of vertices that induces a complete bipartite subgraph of with at least one edge. A clique-colouring of 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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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 .