Computer Science & Engineering: An International Journal (CSEIJ), Vol.2, No.1, February 2012 DOI : 10.5121/cseij.2012.2106 53       Subhashis Banerjee #1 , Debajit Sensarma #1 , Krishnendu Basuli #1 , Saptarshi Naskar #2 and Samar Sen Sarma #3 #1 West Bengal State University, West Bengal, India {mail.sb88,debajit.sensarma2008,krishnendu.basuli}@gmail.com #2 Sarsuna College, West Bengal, India 2 sapgrin@gmail.com #3 University Of Calcutta, West Bengal, India sssarma2001@yahoo.com ABSTRACT The Reconstruction Conjecture has been synthesized under the characterized of matching Polynomial. A Polynomial time algorithm for generating matching polynomial of an undirected graph is given. Algorithms are given for reconstructing a graph from its node-deleted and edge-deleted subgraphs. Also the relation between the isomorphism and reconstruction has been investigated. KEYWORDS Multiset, Card, Deck, Matching, Perfect matching, K-matching, Matching polynomial, Tree- Decomposition, Isomorphism 1. INTRODUCTION The reconstruction conjecture is generally regarded as one of the foremost unsolved problem in graph theory. It was first studied in 1941 by Kelly and Ulam [12]. It got name Reconstruction Conjecture when Harary reformulated it in 1964 [8]. The reconstruction conjecture claims that every graph on at least three vertices is uniquely determined (up to isomorphism) by its collection of vertex deleted subgraphs. Harary[8] formulated the Edge-Reconstruction Conjecture, which states that a finite simple graph with at least four edges can be reconstructed from its collection of one edge deleted sub graphs. The Reconstruction Conjecture is interesting not only from a mathematical or historical point of view but also due to its applicability in diverse fields. Archaeologists may try to assemble broken fragments of pottery to find the shape and pattern of an ancient vase. Chemists may infer the structure of an organic molecule from knowledge of its decomposition products. In bioinformatics the Multiple Sequence- Alignment problem[2] is to reconstruct a sequence with minimum gap insertion and maximum number of matching symbols, given a list of protein or DNA sequences. In computer networking, a reconstruction problem can appear in the following scenario: given a