When Trees Grow Low: Shrubs and Fast MSO1

  • Robert Ganian
  • Petr Hliněný
  • Jaroslav Nešetřil
  • Jan Obdržálek
  • Patrice Ossona de Mendez
  • Reshma Ramadurai
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)


Recent characterization [9] of those graphs for which coloured MSO2 model checking is fast raised the interest in the graph invariant called tree-depth. Looking for a similar characterization for (coloured) MSO1, we introduce the notion of shrub-depth of a graph class. To prove that MSO1 model checking is fast for classes of bounded shrub-depth, we show that shrub-depth exactly characterizes the graph classes having interpretation in coloured trees of bounded height. We also introduce a common extension of cographs and of graphs with bounded shrub-depth — m-partite cographs (still of bounded clique-width), which are well quasi-ordered by the relation “is an induced subgraph of” and therefore allow polynomial time testing of hereditary properties.


Model Check Disjoint Union Rooted Tree Graph Class Graph Interpretation 
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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Robert Ganian
    • 1
  • Petr Hliněný
    • 2
  • Jaroslav Nešetřil
    • 3
  • Jan Obdržálek
    • 2
  • Patrice Ossona de Mendez
    • 4
  • Reshma Ramadurai
    • 2
  1. 1.Institute for InformaticsGoethe UniversityFrankfurtGermany
  2. 2.Faculty of InformaticsMasaryk UniversityBrnoCzech Republic
  3. 3.Computer Science Inst. of Charles University (IUUK)PrahaCzech Republic
  4. 4.École des Hautes Études en Sciences SocialesCNRS UMR 8557ParisFrance

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