Graphs that are almost binary trees

Abstract

This paper studies embeddings of graphs in binary trees. The cost of such an embedding is the maximum distance in the binary tree between images of adjacent graph vertices. Several techniques for bounding the costs of such embeddings from above are derived; notable among these is an algorithm for embedding any outerplanar graph in a binary tree with a cost that is within a factor of 3 of optimal. A number of techniques for bounding the costs of such embeddings from below are developed; notable here are two techniques for inferring the presence of large separators in graphs. Finally, a number of characterizations are established of those families of graphs that are almost binary trees, in the sense that every graph in the family is embeddable in a binary tree within bounded cost.