When Trees Grow Low: Shrubs and Fast MSO1
Recent characterization  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.
Unable to display preview. Download preview PDF.
- 8.Gajarský, J.: Efficient solvability of graph MSO properties. Master’s thesis, Masaryk University, Brno (2012)Google Scholar
- 9.Gajarský, J., Hliněný, P.: Deciding graph MSO properties: Has it all been told already (submitted, 2012)Google Scholar
- 11.Ganian, R., Hliněný, P., Obdržálek, J.: Clique-width: When hard does not mean impossible. In: STACS 2011. LIPIcs, vol. 9, pp. 404–415. Dagstuhl Publishing (2011)Google Scholar
- 16.Nešetřil, J., Ossona de Mendez, P.: Tree-depth, subgraph coloring and homomorphism bounds. European J. Combin. 27(6), 1024–1041 (2006)Google Scholar
- 18.Nešetřil, J., Ossona de Mendez, P.: Sparsity (Graphs, Structures, and Algorithms) Algorithms and Combinatorics, vol. 28, p. 465. Springer (2012)Google Scholar
- 19.Rabin, M.O.: A simple method for undecidability proofs and some applications. In: Bar-Hillel, Y. (ed.) Logic, Methodology and Philosophy of Sciences, vol. 1, pp. 58–68. North-Holland, Amsterdam (1964)Google Scholar
- 20.Schaffer, P.: Optimal node ranking of trees in linear time. Inform. Process. Lett. 33, 91–96 (1989/1990)Google Scholar