On rational approximation of a geometric graph

A geometric graph is rational if all its edges have rational lengths. In 2008 M. Kleber asked for what graph the vertices can be slightly perturbed in their ϵϵ-neighborhoods in such a way that the resulting graph becomes rational (the ϵϵ-approximation) and in addition the vertices can have rational coordinates (the rational ϵϵ-approximation). J. Geelen et al. in 2008 proved that any geometric cubic graph has a rational ϵϵ-approximation for any ϵ>0ϵ>0. In 2011 A. Dubickas assumed the existence of up to four vertices of degree above 3. We prove that any connected geometric graph with maximum degree 4 and a vertex ww of degw<4degw<4 and any 33-tree have ϵϵ-rational approximations for any ϵ>0ϵ>0.