Planarity testing of doubly periodic infinite graphs

This paper describes an efficient way to test the VAP-free (Vertex Accumulation Point free) planarity of oneand two-dimensional dynamic graphs. Dynamic graphs are infinite graphs consisting of an infinite number of basic cells connected regularly according to labels in a finite graph called a sraric graph. Dynamic graphs arize in the design of highly regular VLSI circuits, such as systolic arrays and digital signal processing chips. We show that VAP-free planarity testing of dynamic graphs can be done efficiently by making use of their regularity. First, we will establish necessary conditions for VAP-free planarity of dynamic graphs. Then we show the existence of a small finite graph which is planar if and only if the original dynamic graph is VAP-free planar. From this it follows that VAP-free planarity testing of oneand twodimensional dynamic graphs is asymptomically no more difficult than planarity testing of finite graphs, and thus can be done in linear time.