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 $\textrm{drn}(G)$, is the minimum number of dacards that suffice to reconstruct $G$. The adversary degree-associated reconstruction number $\textrm{adrn}(G)$ is the least $k$ such that every set of $k$ dacards determines $G$. The analogous parameters for degree-associated edge reconstruction are $\textrm{dern}(G)$ and $\textrm{adern}(G)$. We determine these four parameters for all complete multipartite graphs andtheir complements. The answer is usually $2$ for all four parameters,but there are exceptions in each case.