Frontotemporal dementia associated with granulin (GRN) mutations presents asymmetric brain atrophy. We applied a Minimum Spanning Tree plus an Efficiency Cost Optimization approach to cortical thickness data in order to test whether graph theory measures could identify global or local impairment of connectivity in the presymptomatic phase of pathology, where other techniques failed in demonstrating changes. We included 52 symptomatic GRN mutation carriers (SC), 161 presymptomatic GRN mutation carriers (PSC) and 341 non-carriers relatives from the Genetic Frontotemporal dementia research Initiative cohort. Group differences of global, nodal and edge connectivity in (Minimum Spanning Tree plus an Efficiency Cost Optimization) graph were tested via Structural Equation Models. Global graph perturbation was selectively impaired in SC compared to non-carriers, with no changes in PSC. At the local level, only SC exhibited perturbation of frontotemporal nodes, but edge connectivity revealed a characteristic pattern of interhemispheric disconnection, involving homologous parietal regions, in PSC. Our results suggest that GRN-related frontotemporal dementia resembles a disconnection syndrome, with interhemispheric disconnection between parietal regions in presymptomatic phases that progresses to frontotemporal areas as symptoms emerge.