Acyclic, Star and Injective Colouring: A Complexity Picture for H-Free Graphs

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F20%3A10422003" target="_blank" >RIV/00216208:11320/20:10422003 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.4230/LIPIcs.ESA.2020.22" target="_blank" >https://doi.org/10.4230/LIPIcs.ESA.2020.22</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4230/LIPIcs.ESA.2020.22" target="_blank" >10.4230/LIPIcs.ESA.2020.22</a>

Result language

angličtina

Original language name

Acyclic, Star and Injective Colouring: A Complexity Picture for H-Free Graphs

Original language description

A k-colouring c of a graph G is a mapping V(G) RIGHTWARDS ARROW {1,2,... k} such that c(u) NOT EQUAL TO c(v) whenever u and v are adjacent. The corresponding decision problem is Colouring. A colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. Hence, every injective colouring is a star colouring and every star colouring is an acyclic colouring. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring (the last problem is also known as L(1,1)-Labelling). A classical complexity result on Colouring is a well-known dichotomy for H-free graphs, which was established twenty years ago (in this context, a graph is H-free if and only if it does not contain H as an induced subgraph). Moreover, this result has led to a large collection of results, which helped us to better understand the complexity of Colouring. In contrast, there is no systematic study into the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring despite numerous algorithmic and structural results that have appeared over the years. We initiate such a systematic complexity study, and similar to the study of Colouring we use the class of H-free graphs as a testbed. We prove the following results: 1) We give almost complete classifications for the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring for H-free graphs. 2) If the number of colours k is fixed, that is, not part of the input, we give full complexity classifications for each of the three problems for H-free graphs. From our study we conclude that for fixed k the three problems behave in the same way, but this is no longer true if k is part of the input. To obtain several of our results we prove stronger complexity results that in particular involve the girth of a graph and the class of line graphs.

Czech name

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Czech description

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Type

D - Article in proceedings

CEP classification

—

OECD FORD branch

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Project

—

Continuities

S - Specificky vyzkum na vysokych skolach<br>I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2020

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Article name in the collection

28th Annual European Symposium on Algorithms (ESA 2020)

ISBN

978-3-95977-162-7

ISSN

1868-8969

e-ISSN

—

Number of pages

22

Pages from-to

1-22

Publisher name

Leibniz International Proceedings in Informatics (LIPIcs)

Place of publication

Dagstuhl, Německo

Event location

online

Event date

Sep 7, 2020

Type of event by nationality

WRD - Celosvětová akce

UT code for WoS article

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