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.
KeywordsModel Check Disjoint Union Rooted Tree Graph Class Graph Interpretation
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