IWOCA 2012: Combinatorial Algorithms pp 100-109

# Degree Associated Edge Reconstruction Number

• S. Monikandan
• S. Sundar Raj
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7643)

## 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

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