A system $R$ of equivalence relations on a set $A$ (with at least $3$ elements) is \emph{semirigid} if??only the trivial operations??(that is the projections and constant functions) preserve all members of $R$. To a system $R$ of equivalence relations we associate a graph $G_R$. We observe that??if $R$ is semirigid then the graph $G_R$ is $2$-connected. We show that the converse holds if all the members of $R$ are atoms of the lattice $E$ of equivalence relations on $A$. We present a notion of??graphical composition of semirigid systems and show that it preserves semirigidity.