On critical nets in $${\mathbb {R}}^k$$

Critical nets in $${\mathbb {R}}^k$$ (sometimes called geodesic nets) are embedded graph with the property that their embedding is a critical point of total (edge) length functional and under constraint certain 1-valent vertices have fixed position. In contrast to what happens on generic manifolds, we show that, if bounded n number vertices, edges not incident vertex by rn (where r outer radius), degree any (and hence vertices) $$n\ell $$ $$\ell related combinatorial diameter graph).