  Orbits of graphs under local complementation (LC) and edge local
complementation (ELC) have been studied in several different contexts. For
instance, there are connections between orbits of graphs and error-correcting
codes. We define a new graph class, ELC-preserved graphs, comprising all graphs
that have an ELC orbit of size one. Through an exhaustive search, we find all
ELC-preserved graphs of order up to 12 and all ELC-preserved bipartite graphs
of order up to 16. We provide general recursive constructions for infinite
families of ELC-preserved graphs, and show that all known ELC-preserved graphs
arise from these constructions or can be obtained from Hamming codes. We also
prove that certain pairs of ELC-preserved graphs are LC equivalent. We define
ELC-preserved codes as binary linear codes corresponding to bipartite
ELC-preserved graphs, and study the parameters of such codes.
