MPI-INF/SWS Research Reports 1991-2021

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Reachability substitutes for planar digraphs

Katriel, Irit and Kutz, Martin and Skutella, Martin

March 2005, 24 pages.

Status: available - back from printing

Given a digraph $G = (V,E)$ with a set $U$ of vertices marked ``interesting,'' we want to find a smaller digraph $\RS{} = (V',E')$ with $V' \supseteq U$ in such a way that the reachabilities amongst those interesting vertices in $G$ and \RS{} are the same. So with respect to the reachability relations within $U$, the digraph \RS{} is a substitute for $G$. We show that while almost all graphs do not have reachability substitutes smaller than $\Ohmega(|U|^2/\log |U|)$, every planar graph has a reachability substitute of size $\Oh(|U| \log^2 |U|)$. Our result rests on two new structural results for planar dags, a separation procedure and a reachability theorem, which might be of independent interest.

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  AUTHOR = {Katriel, Irit and Kutz, Martin and Skutella, Martin},
  TITLE = {Reachability substitutes for planar digraphs},
  TYPE = {Research Report},
  INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},
  ADDRESS = {Stuhlsatzenhausweg 85, 66123 Saarbr{\"u}cken, Germany},
  NUMBER = {MPI-I-2005-1-002},
  MONTH = {March},
  YEAR = {2005},
  ISSN = {0946-011X},