2-Swappability and the Edge-Reconstruction Number of Regular Graphs

The edge-reconstruction number of graph $G$, denoted $ern(G)$,is the size of the smallest multiset of edge-deleted, unlabeled subgraphs of $G$, from which the structure of $G$ can be uniquely determined. That there was some connection between the areas of edge reconstruction and swappability has been known since the swapping number of a graph was first introduced by Froncek, Rosenberg, and Hlavacek in 2013. This paper illustrates the depth of that connection by proving several bridging results between those areas; in particular, when the graphs in question are both regular and 2-swappable. These connections led to the discovery of four infinite families of $r\geq 3$ regular graphs with $ern(G) \geq 3$, contradicting the formerly conjectured upper bound.