On the reconstraction of the matching polynomial and the reconstruction conjecture

E J Farrell,S A Wahid

doi:10.1155/s016117128700019x

Abstract

Two results are proved. (i) It is shown that the matching polynomial is both node and edge reconstructable. Moreover a practical method of reconstruction is given. (ii) A technique is given for reconstructing a graph from its node‐deleted and edge‐deleted subgraphs. This settles one part of the Reconstruction Conjecture.

Highlights

The graphs considered here will be finite and will have no loops
G, a k is the number of matchings in with k edges and the summation is taken over all the k-matchings in G
Throughout the paper, we will assume that the general graph G has p nodes and q edges unless otherwise specified

Summary

INTRODUCTION

The graphs considered here will be finite and will have no loops. Let G be such a graph. The reconstruction of several graph polynomials has been established, no practical means of reconstruction exists for any of them. Given the deck of the graph G, there is no method available for finding either the characteristic polynomial or the chromatic polynomial of G. Godsil [5] has given a result (Theorem 4.1) which essentially establishes the nodereconstruction of a special form of m(G) His result does not provide a practical reconstruction technique, since he used Tutte’s existence result to establish the reconstruction of the number of perfect matchings in G. The problem of reconstructing a graph from a given deck has always been an interesting one, because of its connection with Ulam’s Conjecture (see [3]). For the graph G, with p nodes and q edges, the node set will be V(G) {v I, v2 v and the edge set, E(G) P

We will denote by

Let v r

Let a k k be