  We evaluate and analyze the exact value of a measure for local genuine
tripartite entanglement in the one-dimensional cluster-Ising model. This model
is attractive since cluster states are considered to be relevant sources for
applying quantum algorithms and the Ising interaction is an expected
perturbation. Whereas bipartite entanglement is identically vanishing, we find
that genuine tripartite entanglement is non zero in the anti-ferromagnetic
phase and also in the cluster phase well before the critical point. We prove
that the measure of local genuine tripartite entanglement captures all the
properties of the topological phase transition. Remarkably, we find that the
amount of genuine tripartite entanglement is independent of whether the
considered ground states satisfy or break the symmetries of the Hamiltonian. We
provide also strong evidences that for this experimentally feasible model local
genuine tripartite entanglement represents the unique non vanishing genuine
multipartite entanglement among any spins.
