Bijective codes for trees and k-trees

The problem of coding labeled trees has been widely studied in the literature and several bijective codes that realize associations between labeled trees and sequences of labels have been presented. k-trees are one of the most natural and interesting generalizations of trees and there is considerable interest in developing efficient tools to manipulate this class of graphs, since many NP-Complete problems have been shown to be polynomially solvable on k-trees and partial k-trees. In 1970 R´enyi and R´enyi generalized the Pr¨ufer code, the first bijective code for trees, to a subset of labeled k-trees. Subsequently, non redundant codes that realize bijection between k-trees (or R´enyi k-trees) and a well defined set of strings were produced. In this paper we introduce a new bijective code for labeled k-trees which, to the best of our knowledge, produces the first coding and decoding algorithms running in linear time with respect to the size of the k-tree.