Degree Associated Edge Reconstruction Parameters of Strong Double Brooms

Abstract

An edge deleted unlabeled subgraph of a graph G is an ecard. A da-ecard specifies the degree of the deleted edge along with the ecard. The degree associated edge reconstruction number of a graph G, dern(G), is the size of the smallest collection of da-ecards of G that uniquely determines G. The adversary degree associated edge reconstruction number of a graph G, adern(G), is the minimum number k such that every collection of k da-ecards of G uniquely determines G. A strong double broom is the graph on at least 5 vertices obtained from a union of (at least two) internally vertex disjoint paths with same ends u and v by appending leaves at u and v. In particular, B(n, n,mPk) is the strong double broom with n leaves at both the ends u and v and with m internally vertex disjoint paths of order k joining u and v. We show that dern of strong double brooms is 1 or 2. We also determine adern(B(n, n,mPk)). It is 3 in most of the cases and 1 or 2 for all the remaining cases, except adern(B(1, 1, 2Pk)) = 5 for k > 3.