Identities for non-crossing graphs and multigraphs

We refine an identity between the numbers of certain non-crossing graphs and multigraphs, by modifying a bijection found by P. Podbrdský [A bijective proof of an identity for noncrossing graphs, Discrete Math. 260 (2003) 249–253]. We also prove a new identity between the number of acyclic non-crossing graphs with image vertices and image edges (isolated vertices allowed and no multiple edges allowed), and the number of non-crossing connected graphs with image edges and image vertices (multiple edges allowed and no isolated vertices allowed).