eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-07-07
63:1
63:14
10.4230/LIPIcs.ICALP.2017.63
article
Neighborhood Complexity and Kernelization for Nowhere Dense Classes of Graphs
Eickmeyer, Kord
Giannopoulou, Archontia C.
Kreutzer, Stephan
Kwon, O-joung
Pilipczuk, Michal
Rabinovich, Roman
Siebertz, Sebastian
We prove that whenever G is a graph from a nowhere dense graph class C, and A is a subset of vertices of G, then the number of subsets of A that are realized as intersections of A with r-neighborhoods of vertices of G is at most f(r,eps)|A|^(1+eps), where r is any positive integer, eps is any positive real, and f is a function that depends only on the class C. This yields a characterization of nowhere dense classes of graphs in terms of neighborhood complexity, which answers a question posed by [Reidl et al., CoRR, 2016]. As an algorithmic application of the above result, we show that for every fixed integer r, the parameterized Distance-r Dominating Set problem admits an almost linear kernel on any nowhere dense graph class. This proves a conjecture posed by [Drange et al., STACS 2016], and shows that the limit of parameterized tractability of Distance-r Dominating Set on subgraph-closed graph classes lies exactly on the boundary between nowhere denseness and somewhere denseness.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol080-icalp2017/LIPIcs.ICALP.2017.63/LIPIcs.ICALP.2017.63.pdf
Graph Structure Theory
Nowhere Dense Graphs
Parameterized Complexity
Kernelization
Dominating Set