2-Swappability and the Edge-Reconstruction Number of Regular Graphs
Abstract
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≥ 3 regular graphs with ern(G) ≥ 3, contradicting the formerly conjectured upper bound.
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