  In \cite{4} Kauffman and Vogel constructed a rigid vertex regular isotopy
invariant for unoriented four-valent graphs embedded in three dimensional
space. It assigns to each embedded graph $G$ a polynomial, denoted $[G]$, in
three variables, $A$, $B$ and $a$, satisfies the skein relation: $$
[\psdiag{2}{6}{overcross}]=A [\psdiag{2}{6}{horlines}]+B
[\psdiag{2}{6}{verlines}]+[\psdiag{2}{6}{vertex}]$$ and is defined in terms of
a state-sum and the Dubrovnik polynomial for links. Using the graphical
calculus of \cite{4} it is shown that the polynomial of a planar graph can be
calculated recursively from that of planar graphs with less vertices, which
also allows the polynomial of an embedded graph to be calculated without
resorting to links. The same approach is used to give a direct proof of
uniqueness of the (normalized) polynomial restricted to planar graphs. In the
case $B=A^{-1}$ and $a=A$, it is proved that for a planar graph $G$ we have
$[G]=2^{c-1}(-A-A^{-1})^v $, where $c$ is the number of connected components of
$G$ and $v$ is the number of vertices of $G$. As a corollary, a necessary, but
not sufficient, condition is obtained for an embedded graph to be ambient
isotopic to a planar graph.
