Traversal with Enumeration of Geometric Graphs in Bounded Space

Abstract

In this paper, we provide an algorithm for traversing geometric graphs which visits all vertices, and reports every vertex and edge exactly once. To achieve this, we combine a given geometric graph $G$ with the integer lattice, seen as a graph, in such a way that the resulting hypothetical graph can be traversed using the algorithm in \cite{Chavez}. To overcome the problem with hypothetical vertices and edges, we develop an algorithm for visiting any $k$th neighborhood of a vertex in a graph straight-line drawn in the plane using $O(\log k)$ memory. The memory needed to complete the traversal of a geometric graph then turns out to depend on the maximum ratio of the graph distance and Euclidean distance for pairs of distinct vertices of $G$ at Euclidean distance greater than one and less than $2\sqrt{2}$.