Abstract
We investigate the parameterized complexity of the recognition problem for the proper Hgraphs. The Hgraphs are the intersection graphs of connected subgraphs of a subdivision of a multigraph H, and the properness means that the containment relationship between the representations of the vertices is forbidden. The class of Hgraphs was introduced as a natural (parameterized) generalization of interval and circulararc graphs by Biró, Hujter, and Tuza in 1992, and the proper Hgraphs were introduced by Chaplick et al. in WADS 2019 as a generalization of proper interval and circulararc graphs. For these graph classes, H may be seen as a structural parameter reflecting the distance of a graph to a (proper) interval graph, and as such gained attention as a structural parameter in the design of efficient algorithms. We show the following results.
 For a tree T with t nodes, it can be decided in 2^{𝒪(t² log t)} ⋅ n³ time, whether an nvertex graph G is a proper Tgraph. For yesinstances, our algorithm outputs a proper Trepresentation. This proves that the recognition problem for proper Hgraphs, where H required to be a tree, is fixedparameter tractable when parameterized by the size of T. Previously only NPcompleteness was known.
 Contrasting to the first result, we prove that if H is not constrained to be a tree, then the recognition problem becomes much harder. Namely, we show that there is a multigraph H with 4 vertices and 5 edges such that it is NPcomplete to decide whether G is a proper Hgraph.
BibTeX  Entry
@InProceedings{chaplick_et_al:LIPIcs:2020:13311,
author = {Steven Chaplick and Petr A. Golovach and Tim A. Hartmann and Du{\v{s}}an Knop},
title = {{Recognizing Proper TreeGraphs}},
booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)},
pages = {8:18:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771726},
ISSN = {18688969},
year = {2020},
volume = {180},
editor = {Yixin Cao and Marcin Pilipczuk},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/13311},
URN = {urn:nbn:de:0030drops133118},
doi = {10.4230/LIPIcs.IPEC.2020.8},
annote = {Keywords: intersection graphs, Hgraphs, recognition, fixedparameter tractability}
}
Keywords: 

intersection graphs, Hgraphs, recognition, fixedparameter tractability 
Collection: 

15th International Symposium on Parameterized and Exact Computation (IPEC 2020) 
Issue Date: 

2020 
Date of publication: 

04.12.2020 