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 ϵ -neighborhoods in such a way that the resulting graph becomes rational (the ϵ -approximation) and in addition the vertices can have rational coordinates (the rational ϵ -approximation). J. Geelen et al. in 2008 proved that any geometric cubic graph has a rational ϵ -approximation for any ϵ > 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 w of deg w < 4 and any 3 -tree have ϵ -rational approximations for any ϵ > 0 .