Topological graph dimension

In the invited chapter Discrete Spatial Models of the book Handbook of Spatial Logics, we have introduced the concept of dimension for graphs, which is inspired by Evako’s idea of dimension of graphs [A.V. Evako, R. Kopperman, Y.V. Mukhin, Dimensional properties of graphs and digital spaces, J. Math. Imaging Vision 6 (1996) 109–119]. Our definition is analogous to that of (small inductive) dimension in topology. Besides the expected properties of isomorphism-invariance and monotonicity with respect to subgraph inclusion, it has the following distinctive features:
• Local aspect. That is, dimension at a vertex is basic, and the dimension of a graph is obtained as the sup over its vertices.
• Dimension of a strong product G×HG×H is dim(G)+dim(H)dim(G)+dim(H) (for non-empty graphs G,HG,H). In this paper we present a short account of the basic theory, with several new applications and results.