{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article13025","name":"Parameterized Complexity of Edge-Coloured and Signed Graph Homomorphism Problems","abstract":"We study the complexity of graph modification problems with respect to homomorphism-based colouring properties of edge-coloured graphs. A homomorphism from an edge-coloured graph G to an edge-coloured graph H is a vertex-mapping from G to H that preserves adjacencies and edge-colours. We consider the property of having a homomorphism to a fixed edge-coloured graph H, which generalises the classic vertex-colourability property. The question we are interested in is the following: given an edge-coloured graph G, can we perform k graph operations so that the resulting graph admits a homomorphism to H? The operations we consider are vertex-deletion, edge-deletion and switching (an operation that permutes the colours of the edges incident to a given vertex). Switching plays an important role in the theory of signed graphs, that are 2-edge-coloured graphs whose colours are the signs + and -. We denote the corresponding problems (parameterized by k) by Vertex Deletion-H-Colouring, Edge Deletion-H-Colouring and Switching-H-Colouring. These problems generalise the extensively studied H-Colouring problem (where one has to decide if an input graph admits a homomorphism to a fixed target H). For 2-edge-coloured H, it is known that H-Colouring already captures the complexity of all fixed-target Constraint Satisfaction Problems.\r\nOur main focus is on the case where H is an edge-coloured graph of order at most 2, a case that is already interesting since it includes standard problems such as Vertex Cover, Odd Cycle Transversal and Edge Bipartization. For such a graph H, we give a PTime\/NP-complete complexity dichotomy for all three Vertex Deletion-H-Colouring, Edge Deletion-H-Colouring and Switching-H-Colouring problems. Then, we address their parameterized complexity. We show that all Vertex Deletion-H-Colouring and Edge Deletion-H-Colouring problems for such H are FPT. This is in contrast with the fact that already for some H of order 3, unless PTime = NP, none of the three considered problems is in XP, since 3-Colouring is NP-complete. We show that the situation is different for Switching-H-Colouring: there are three 2-edge-coloured graphs H of order 2 for which Switching-H-Colouring is W[1]-hard, and assuming the ETH, admits no algorithm in time f(k)n^{o(k)} for inputs of size n and for any computable function f. For the other cases, Switching-H-Colouring is FPT.","keywords":["Graph homomorphism","Graph modification","Edge-coloured graph","Signed graph"],"author":[{"@type":"Person","name":"Foucaud, Florent","givenName":"Florent","familyName":"Foucaud","affiliation":"Univ. Orl\u00e9ans, INSA Centre Val de Loire, LIFO EA 4022, F-45067 Orl\u00e9ans Cedex 2, France"},{"@type":"Person","name":"Hocquard, Herv\u00e9","givenName":"Herv\u00e9","familyName":"Hocquard","affiliation":"Univ. Bordeaux, Bordeaux INP, CNRS, LaBRI, UMR5800, F-33400 Talence, France"},{"@type":"Person","name":"Lajou, Dimitri","givenName":"Dimitri","familyName":"Lajou","affiliation":"Univ. Bordeaux, Bordeaux INP, CNRS, LaBRI, UMR5800, F-33400 Talence, France"},{"@type":"Person","name":"Mitsou, Valia","givenName":"Valia","familyName":"Mitsou","affiliation":"Universit\u00e9 Paris-Diderot, IRIF, CNRS, 75205, Paris, France"},{"@type":"Person","name":"Pierron, Th\u00e9o","givenName":"Th\u00e9o","familyName":"Pierron","affiliation":"Univ. Bordeaux, Bordeaux INP, CNRS, LaBRI, UMR5800, F-33400 Talence, France"}],"position":15,"pageStart":"15:1","pageEnd":"15:16","dateCreated":"2019-12-04","datePublished":"2019-12-04","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Foucaud, Florent","givenName":"Florent","familyName":"Foucaud","affiliation":"Univ. Orl\u00e9ans, INSA Centre Val de Loire, LIFO EA 4022, F-45067 Orl\u00e9ans Cedex 2, France"},{"@type":"Person","name":"Hocquard, Herv\u00e9","givenName":"Herv\u00e9","familyName":"Hocquard","affiliation":"Univ. Bordeaux, Bordeaux INP, CNRS, LaBRI, UMR5800, F-33400 Talence, France"},{"@type":"Person","name":"Lajou, Dimitri","givenName":"Dimitri","familyName":"Lajou","affiliation":"Univ. Bordeaux, Bordeaux INP, CNRS, LaBRI, UMR5800, F-33400 Talence, France"},{"@type":"Person","name":"Mitsou, Valia","givenName":"Valia","familyName":"Mitsou","affiliation":"Universit\u00e9 Paris-Diderot, IRIF, CNRS, 75205, Paris, France"},{"@type":"Person","name":"Pierron, Th\u00e9o","givenName":"Th\u00e9o","familyName":"Pierron","affiliation":"Univ. Bordeaux, Bordeaux INP, CNRS, LaBRI, UMR5800, F-33400 Talence, France"}],"copyrightYear":"2019","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.IPEC.2019.15","funding":"This research was financed by the ANR project HOSIGRA (ANR-17-CE40-0022) and the IFCAM project \"Applications of graph homomorphisms\" (MA\/IFCAM\/18\/39).","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6351","volumeNumber":148,"name":"14th International Symposium on Parameterized and Exact Computation (IPEC 2019)","dateCreated":"2019-12-04","datePublished":"2019-12-04","editor":[{"@type":"Person","name":"Jansen, Bart M. P.","givenName":"Bart M. P.","familyName":"Jansen","email":"mailto:B.M.P.Jansen@tue.nl","sameAs":"https:\/\/orcid.org\/0000-0001-8204-1268","affiliation":"Eindhoven University of Technology, the Netherlands"},{"@type":"Person","name":"Telle, Jan Arne","givenName":"Jan Arne","familyName":"Telle","email":"mailto:Jan.Arne.Telle@uib.no","sameAs":"https:\/\/orcid.org\/0000-0002-9429-5377","affiliation":"University of Bergen, Norway"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article13025","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6351"}}}