The degree-associated edge-reconstruction number of disconnected graphs and trees

TL;DR

This paper explores how the degree-associated edge-reconstruction number differs from the edge-reconstruction number in disconnected graphs and trees, providing new insights and conjectures on their relationships.

Contribution

It investigates the variation between dern(G) and ern(G) for disconnected graphs and trees, and proposes a conjecture that dern(T) is at most 2 for any tree T.

Findings

Two da-ecards can uniquely identify caterpillars.

Conjecture: dern(T) <= 2 for any tree T.

Known results on edge-reconstruction numbers are summarized.

Abstract

An edge-card of a graph G is a subgraph formed by deleting an edge. The edge-reconstruction number of a graph G, ern(G), is the minimum number of edge-cards required to determine G up to isomorphism. A da-ecard is an edge-card which also specifies the degree of the deleted edge, that is, the number of edges adjacent to it. The degree-associated edge-reconstruction number, dern(G) is the minimum number of da- ecards that suffice to determine the graph G. In this paper we state some known results on the edge-reconstruction number of disconnected graphs and trees. Then we investigate how the degree-associated edge- reconstruction number of disconnected graphs and trees vary from their respective edge-reconstruction number. We show how we can select two da-ecards to identify caterpillars uniquely. Finally we conjecture that for any tree T, dern(T)<= 2.