  Motivated by his studies in knot theory V. Vassiliev introduced $X$-graphs as
regular 4-valent graph with a structure of pairs of opposite edges at each
vertex. He conjectured the conditions under which $X$-graph can be embedded
into a plane respecting the the $X$-structure at every vertex. The conjecture
was proved by V.Manturov. Here we generalize these results for graphs with
vertices of valency 4 or 6, *-graphs. A problem of such generalization was
posted by A.Skopenkov.
