The graph reconstruction number

Abstract

Dedicated ro the memory of Stan Ulam (31. ABSTRACTThe reconstruction number of graph G is the minimum number of pointdeleted subgraphs required in order to uniquely identify the original graph G. We list, based on computer calculations, the reconstruction number for all graphs with at most seven points. Some constructions and conjectures for graphs of higher order are given. The most striking statement is our concluding conjecture that almost all graphs have reconstruction number three.

Degree Associated Edge Reconstruction Number of Graphs with Regular Pruned Graph

“…A vertex deleted subgraph or a card G − v of a graph G is the unlabeled graph obtained from G by deleting a vertex v and all edges incident with v. The ordered pair (d(v), G−v) is called a degree associated card or dacard of the graph G, where d(v) is the degree of v in G. The deck (dadeck) of a graph G is the collection of all its cards (dacards). Following the formulation in [3], a graph G is reconstructible if it can be uniquely determined from its dadeck. For a reconstructible graph G, Harary and Plantholt [3] defined the reconstruction number, rn(G), to be the minimum number of cards which can only belong to its deck and not in the deck of any other graph H, H G, thus uniquely identifying G. An s-blocking set of G is a family F of graphs not isomorphic to G such that every collection of s cards of G will appear in the deck of some graph of F and every graph in F will have s cards in common with G. For a reconstructible graph G, Myrvold [7] studied, the adversary reconstruction number, which is the minimum number k such that every collection of k cards of G is not contained in the deck of any other graph H, H ∼ = G. A graph non-isomorphic to G but having s cards in common with G is called an s-adversary-blocking graph of G. For a reconstructible graph G from its dadeck, Ramachandran [8] defined that the degree associated reconstruction number, drn(G), is the minimum number of dacards that uniquely determines G. The adversary degree associated reconstruction number of a graph G, adrn(G), is the minimum number k such that every collection of k dacards of G uniquely determines G. The degree of an edge e, denoted by d(e), is the number of edges adjacent to e. The edge reconstruction number, degree associated edge reconstruction number and adversary degree associated edge reconstruction number of a graph are defined analogously with edge deletions instead of vertex deletions.…”

Degree-Associated Reconstruction Parameters of Complete Multipartite Graphs and Their Complements

“…For a reconstructible graph G, Harary and Plantholt [4] introduced the reconstruction number, denoted by rn(G); it is the least k such that some multiset of k cards from the deck of G determines G (meaning that every graph not isomorphic to G shares at most rn(G) − 1 dern(G) is the minimum k such that some multiset of k decards determines G, and adern(G) is the minimum k such that every multiset of k decards determines G.…”

A Note on the Adversary Degree Associated Reconstruction Number of Graphs

“…For a reconstructible graph , Harary and Plantholt [3] have defined the reconstruction number ( ) to be the size of the smallest subcollection of the deck of which is not contained in the deck of any other graph ; ≇ . Myrvold [4] referred to this number as ally-reconstruction number of .…”

“…The following weakening of the reconstruction problem has also been considered by Harary and Plantholt [3]. A graph , in a given class of graphs C, is called class-reconstructible if whenever ∈ C has the same deck as , then ≅ .…”