Vertex identification in trees

A red-white coloring of a nontrivial connected graph G of diameter d is an assignment of red and white colors to the vertices of G where at least one vertex is colored red. Associated with each vertex v of G is a d-vector, called the code of v, whose ith coordinate is the number of red vertices at distance i from v. A red-white coloring of G for which distinct vertices have distinct codes is called an identification coloring or ID-coloring of G. A graph G possessing an ID-coloring is an ID-graph. The minimum number of red vertices among all ID-colorings of an ID-graph G is the identification number or ID-number of G. Necessary conditions are established for those trees that are ID-graphs. A tree T is starlike if T is obtained by subdividing the edges of a star of order 4 or more. It is shown that for every positive integer r different from 2, there exist starlike trees satisfying some prescribed properties having ID-number r.