We study the notion of k-stabilizer universal quantum state, that is, an n-qubit quantum state, such that it is possible to induce any stabilizer state on any k qubits, by using only local operations and classical communications. These states generalize the notion of k-pairable states introduced by Bravyi et al., and can be studied from a combinatorial perspective using graph states and k-vertex-minor universal graphs. First, we demonstrate the existence of k-stabilizer universal graph states that are optimal in size with n = Θ(k2) qubits. We also provide parameters for which a random graph state on Θ(k2) qubits is k-stabilizer universal with high probability. Our second contribution consists of two explicit constructions of k-stabilizer universal graph states on n = O(k 4) qubits. Both rely upon the incidence graph of the projective plane over a finite field Fq. This provides a major improvement over the previously known explicit construction of k-pairable graph states with n = O(23k), bringing forth a new and potentially powerful family of multipartite quantum resources.