Logical complexity of graphs: a survey

We discuss the definability of finite graphs in first-order logic with two relation symbols for adjacency and equality of vertices. The logical depth D(G) of a graph G is equal to the minimum quantifier depth of a sentence defining G up to isomorphism. The logical width W (G) is the minimum number of variables occurring in such a sentence. The logical length L(G) is the length of a shortest defining sentence. We survey known estimates for these graph parameters and discuss their relations to other topics (such as the efficiency of the Weisfeiler-Lehman algorithm in isomorphism testing, the evolution of a random graph, quantitative characteristics of the zero-one law, or the contribution of Frank Ramsey to the research on Hilbert’s Entscheidungsproblem). Also, we trace the behavior of the descriptive complexity of a graph as the logic becomes more restrictive (for example, only definitions with a bounded number of variables or quantifier alternations are allowed) or more expressible (after powering with counting quantifiers). Date: 8 January 2011. ∗ Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA. This work done under the support of the National Science Foundation (Grant DMS-0758057) and the Alexander von Humboldt Foundation. † Institute for Applied Problems of Mechanics and Mathematics, 79060 Lviv, Ukraine. This work was done under the support of the Alexander von Humboldt Foundation.