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We investigate the complexity and expressive power of a spatial logic for reasoning about graphs. This logic was previously introduced by Cardelli, Gardner and Ghelli, and provides the simplest setting in which to explore such results for spatial logics. We study several forms of the logic: the logic with and without recursion, and with either an exponential or a linear version of the basic composition operator. We study the combined complexity and the expressive power of the four combinations. We prove that, without recursion, the linear and exponential versions of the logic correspond to significant fragments of first-order (FO) and monadic second-order (MSO) Logics; the two versions are actually equivalent to FO and MSO on graphs representing strings. However, when the two versions are enriched with μ-style recursion, their expressive power is sharply increased.Both are able to express PSPACE-complete …