# 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 *K* _{1,3}. 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.

## Keywords

reconstruction number edge reconstruction number card dacard## Preview

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