Degree Associated Edge Reconstruction Parameters of Strong Double Brooms

TL;DR

This paper investigates the degree associated edge reconstruction parameters of strong double broom graphs, determining their minimal collection sizes needed for unique reconstruction and revealing specific values for various configurations.

Contribution

It introduces the concepts of dern and adern for strong double brooms and provides exact values for these parameters across different graph structures.

Findings

dern of strong double brooms is 1 or 2

adern is mostly 3, sometimes 1 or 2

adern for B(1, 1, 2Pk) is 5 for k > 3

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.