  Let ${\cal{C}}_1$ be the set of fundamental cycles of breadth-first-search
trees in a graph $G$ and ${\cal{C}}_2$ the set of the sums of two cycles in
${\cal{C}}_1$. Then we show that $(1) {\cal{C}}={\cal{C}}_1\bigcup{\cal{C}}_2$
contains a shortest $\Pi$-twosided cycle in a $\Pi$-embedded graph $G$;$(2)$
$\cal{C}$ contains all the possible shortest even cycles in a graph $G$;$(3)$
If a shortest cycle in a graph $G$ is an odd cycle, then $\cal{C}$ contains all
the shortest odd cycles in $G$. This implies the existence of a polynomially
bounded algorithm to find a shortest $\Pi-$twosided cycle in an embedded graph
and thus solves an open problem of B.Mohar and C.Thomassen[2,pp112]
