Degree-associated reconstruction parameters of complete multipartite graphs and their complements

Abstract

A vertex-deleted subgraph of a graph G is a card. A dacard consists of a card and the degree of the missing vertex. The degree-associated reconstruction number of a graph G, denoted drn(G), is the minimum number of dacards that suffice to reconstruct G. The adversary degree-associated reconstruction number adrn(G) is the least k such that every set of k dacards determines G. The analogous parameters for degree-associated edge reconstruction are dern(G) and adern(G). We determine these four parameters for all complete multipartite graphs and their complements. The answer is usually 2 for all four parameters, but there are exceptions in each case.