  Suppose that you add rigid bars between points in the plane, and suppose that
a constant fraction $q$ of the points moves freely in the whole plane; the
remaining fraction is constrained to move on fixed lines called sliders. When
does a giant rigid cluster emerge? Under a genericity condition, the answer
only depends on the graph formed by the points (vertices) and the bars (edges).
We find for the random graph $G \in \mathcal{G}(n,c/n)$ the threshold value of
$c$ for the appearance of a linear-sized rigid component as a function of $q$,
generalizing results of Kasiviswanathan et al. We show that this appearance of
a giant component undergoes a continuous transition for $q \leq 1/2$ and a
discontinuous transition for $q > 1/2$. In our proofs, we introduce a
generalized notion of orientability interpolating between 1- and
2-orientability, of cores interpolating between 2-core and 3-core, and of
extended cores interpolating between 2+1-core and 3+2-core; we find the precise
expressions for the respective thresholds and the sizes of the different cores
above the threshold. In particular, this proves a conjecture of Kasiviswanathan
et al. about the size of the 3+2-core. We also derive some structural
properties of rigidity with sliders (matroid and decomposition into components)
which can be of independent interest.
