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In this paper we study the recently introduced "shared reward dilemma" (Cuesta et al. in J Theor Biol 251:253&-263, 2008) in the presence of a structure governing the interactions among the population. The shared reward dilemma arises when the prisoner's dilemma is supplemented with a second stage in which a fixed reward is equally distributed among all cooperators. We first extend our previous results on the equilibrium structure of this game to the case of a one-shot game taking place on a regular network. Subsequently, we consider an evolutionary version of the game on both lattices and random networks. We show that the evolutionary game on graphs exhibits important differences with the case of well-mixed populations. In particular, there exists an important parameter range in which the cooperation is boosted and a single cooperator can invade a population of defectors. We study the dependence of the cooperation levels on the neighborhood size, finding that on random networks the level of cooperation reached decreases as the neighborhood size increases. Moreover, square lattices favor cooperation more than random networks, and on them cooperation may be almost full for certain parameter regions even for large neighborhood sizes. Further, we show that the effect of the population structure is never detrimental for cooperation.We interpret our results in terms of weak versus strong temptation and discuss the nontrivial issues involved in trying to promote cooperation exogenously by means of such a reward mechanism.