Degree-associated edge-reconstruction numbers of double-brooms

TL;DR

This paper determines the degree-associated edge-reconstruction numbers for all double-brooms, showing that usually one or two decards suffice, with some notable exceptions.

Contribution

It provides the first complete characterization of degree-associated edge-reconstruction numbers for double-brooms, including identifying exceptions.

Findings

$ ext{dern}(G)$ is usually 1 for double-brooms.

$ ext{adern}(G)$ is usually 2 for double-brooms.

There are specific exceptions to these typical values.

Abstract

An edge-deleted subgraph of a graph $G$ is an {\it edge-card}. A {\it decard} consists of an edge-card and the degree of the missing edge. The {\it degree-associated edge-reconstruction number} of a graph $G$, denoted $\dern(G)$, is the minimum number of decards that suffice to reconstruct $G$. The {\it adversary degree-associated edge-reconstruction number} $\adern(G)$ is the least $k$ such that every set of $k$ decards determines $G$. We determine these two parameters for all double-brooms. The answer is usually $1$ for $\dern(G)$, and $2$ for $\adern(G)$ when $G$ is double-broom. But there are exceptions in each case.