Degree associated edge reconstruction number

Abstract

An edge-deleted subgraph of a graph G is called an ecard of G. An ecard of G with which the degree of the deleted edge is also given is called a degree associated ecard (or da-ecard) of G. The edeck (da-edeck) of a graph G is its collection of ecards (da-ecards). The degree associated edge reconstruction number, dern(G), of a graph G is the size of the smallest collection of ecards of G uniquely determines G. The adversary degree associated edge reconstruction number, adern(G), of a graph G is the minimum number k such that every collection of k da-ecards of G uniquely determines G. We prove that dern(G)= adern(G)=1 for any regular graph G or any bidegreed graph G with exactly one vertex of different degree, which differs by at least three. We determine dern and adern for all complete bipartite graphs except K1,3. We also prove that dern(G) ≤ 2 and adern(G) ≤ 3 for any complete 3-partite graph G with n vertices in which all partite sets are equal in size as possible and a few other results. © 2012 Springer-Verlag.