Reconstructibility of Tree Graphs

Abstract

We present a polynomial representation of the STF-vector (subtree size frequencies) that we use for creating algorithms to calculate the vector from free trees. We show that there exist examples of non-isomorphic pairs of unlabeled free trees that are STF-equivalent. Using exhaustive computer search, we determine all such pairs for small sizes. We show that there are infinitely many non-isomorphic STF-equivalent pairs of trees by constructing infinite families of examples. We present reconstructibility results for special kinds of trees. We investigate when tree T can be reconstructed from the multiset of Parikh vectors of all of its subtrees, or all of its paths, or all of its maximal paths. We present algorithms to calculate the multiset of Parikh vectors based on subree, path and maximal path information for unlabeled, labeled and weighted cases. We present infinite families of non-isomorphic equivalent (multisets of Parikh vector) trees for subtree, path and maximal path information in unlabeled, labeled and weighted cases. We also determined all such pairs for small sizes using exhaustive computer search. We present reconstruction algorithms and non-reconstructibility results, and extend the polynomial method, previously applied to jumbled and weighted strings. We present a polynomial representation of the RSTF-vector (rooted subtree size frequencies) and well-formed polynomial representation of the vector that we use for creating algorithms to calculate the vector from free trees. We show that there exist examples of non-isomorphic pairs of rooted directed trees that are RSTF-equivalent. Using exhaustive computer search, we determine all such pairs for small sizes. We present infinite families of RSTF-equivalent rooted directed trees and show a method to create such families from any non-isomorphic rooted directed RSTF-equivalent trees. We present the RRDT algorithm for reconstructing rooted directed trees from their RSTF-polynomial.