Easy problems for grid-structured graphs

Abstract

This article concentrates on classes of graphs containing large grids and having a very regular structure. Grid-structured hierarchical graphs are defined in [19] by giving a static graph defining the content of a cell of a ddimensional grid, repeating this static graph in each cell and by connecting the vertices in cells of a local neighborhood corresponding to a finite transit function in a uniform way. It is shown that for each finitely represented class K of dynamic graphs all monotone graph properties and all first order (FO) problems can be solved in constant time O(1). This result improves the linear time computability of FO problems for graphs of bounded degree from [25]. © Springer-Verlag Berlin Heidelberg 2007.