When Trees Grow Low: Shrubs and Fast MSO1
- Cite this paper as:
- Ganian R., Hliněný P., Nešetřil J., Obdržálek J., Ossona de Mendez P., Ramadurai R. (2012) When Trees Grow Low: Shrubs and Fast MSO1. In: Rovan B., Sassone V., Widmayer P. (eds) Mathematical Foundations of Computer Science 2012. MFCS 2012. Lecture Notes in Computer Science, vol 7464. Springer, Berlin, Heidelberg
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.
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