Reconstruction Conjecture for Graphs Isomorphic to Cube of a Tree

TL;DR

This paper proves the reconstruction conjecture for graphs that are isomorphic to the cube of a tree, using properties of trees and their peripheral vertex deleted subgraphs.

Contribution

It establishes the reconstruction conjecture for the class of graphs isomorphic to the cube of a tree, introducing new characterizations and recognizability results.

Findings

Reconstruction conjecture holds for graphs isomorphic to the cube of a tree.

Characterization and recognizability of the cube of a tree are established.

Uniqueness of the original tree as a cube root is proven, except for complete graphs.

Abstract

This paper proves the reconstruction conjecture for graphs which are isomorphic to the cube of a tree. The proof uses the reconstructibility of trees from their peripheral vertex deleted subgraphs. The main result follows from (i) characterization of the cube of a tree (ii) recognizability of the cube of a tree (iii) uniqueness of tree as a cube root of a graph G, except when G is a complete graph (iv) reconstructibility of trees from their peripheral vertex deleted subgraphs.