eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-01
11:1
11:16
10.4230/LIPIcs.SoCG.2022.11
article
True Contraction Decomposition and Almost ETH-Tight Bipartization for Unit-Disk Graphs
Bandyapadhyay, Sayan
1
Lochet, William
2
Lokshtanov, Daniel
3
Saurabh, Saket
4
Xue, Jie
5
University of Bergen, Norway
LIRMM, UniversitΓ© de Montpellier, CNRS, France
University of California, Santa Barbara, CA, USA
Institute of Mathematical Sciences, Chennai, India
New York University Shanghai, China
We prove a structural theorem for unit-disk graphs, which (roughly) states that given a set π of n unit disks inducing a unit-disk graph G_π and a number p β [n], one can partition π into p subsets πβ,β¦ ,π_p such that for every i β [p] and every π' β π_i, the graph obtained from G_π by contracting all edges between the vertices in π_i $1π' admits a tree decomposition in which each bag consists of O(p+|π'|) cliques. Our theorem can be viewed as an analog for unit-disk graphs of the structural theorems for planar graphs and almost-embeddable graphs proved very recently by Marx et al. [SODA'22] and Bandyapadhyay et al. [SODA'22].
By applying our structural theorem, we give several new combinatorial and algorithmic results for unit-disk graphs. On the combinatorial side, we obtain the first Contraction Decomposition Theorem (CDT) for unit-disk graphs, resolving an open question in the work Panolan et al. [SODA'19]. On the algorithmic side, we obtain a new FPT algorithm for bipartization (also known as odd cycle transversal) on unit-disk graphs, which runs in 2^{O(βk log k)} β
n^{O(1)} time, where k denotes the solution size. Our algorithm significantly improves the previous slightly subexponential-time algorithm given by Lokshtanov et al. [SODA'22] (which works more generally for disk graphs) and is almost optimal, as the problem cannot be solved in 2^{o(βk)} β
n^{O(1)} time assuming the ETH.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol224-socg2022/LIPIcs.SoCG.2022.11/LIPIcs.SoCG.2022.11.pdf
unit-disk graphs
tree decomposition
contraction decomposition
bipartization