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https://doi.org/10.1007/978-1-4842-6603-8irst-order formulae cannot have a global view of a structure. This limitation ensures their low sequential computational complexity. We show that the locality impacts as well on their distributed computational complexity. We use first-order formulae to describe the properties of finite connected gra

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https://doi.org/10.1057/9780230625198duced prime graphs with respect to . and an undirected graph has rank-width at most 1 if and only if it is a distance-hereditary graph. We are interested in an extension of these results to directed graphs. We give several characterizations of directed graphs of rank-width 1 and we prove that the ra

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https://doi.org/10.1007/978-3-642-76401-1 with respect to the number of vertices. We also show that its running time is 2.1364... when the goal is to find a spanning tree with at least . internal vertices. Both running time bounds are obtained via a Measure & Conquer analysis, the latter one being a novel use of this kind of analysis for parameterized algorithms.

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Mathematics and Its Applications of size . (and at most .) are W[1]-complete problems (when parameterized by .) for any pair of finite sets . and .. We further present results on dual parametrization by . − ., and results on certain infinite sets (in particular for ., . being the sets of even and odd integers).

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