Causal graphs may inform covariate adjustment for estimating causal effects and improve estimation efficiency by exploiting the graphical structure. In many applications, however, the target causal parameter may not be point-identified due to the presence of unmeasured confounding. Sensitivity analysis methods address this challenge by characterizing bounds on the causal parameter under varying assumptions about the magnitude or form of unmeasured confounding. We focus on semiparametric efficient estimation of causal effects in non-identifiable settings, assuming a known (or hypothesized) causal graph. We propose an influence function projection approach that exploits the conditional independence constraints implied by the graph to improve the efficiency of semiparametric estimators of upper and lower bounds on the average causal effect under a given sensitivity analysis model. Our approach applies across multiple sensitivity analysis frameworks and causal estimands, thereby connecting knowledge of graphical structure with the sensitivity analysis literature. We illustrate our approach through simulations and real data examples thought to be affected by unmeasured confounding, including the effect of labor training program on post-intervention earnings, and the effect of low ejection fraction on heart failure death.

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