## Introduction

Unlike traditional kin selection or inclusive fitness theory (Hamilton, 1964), which emphasizes how relatedness between individuals promotes helping behaviour, evolutionary graph theory emphasizes the importance of the spatial subdivision of populations (Santos & Pacheco, 2005; Nowak, 2006; Ohtsuki *et al.*, 2006; Ohtsuki & Nowak, 2006). Evolutionary graph theory models allow the effect of space on helping to be investigated in a broad context, while simultaneously making the models mathematically tractable (Lieberman *et al.*, 2005; Ohtsuki *et al.*, 2006; Ohtsuki & Nowak, 2006). Recently, a rule has been derived (Nowak, 2006; Ohtsuki *et al.*, 2006; Ohtsuki & Nowak, 2006) for the evolution of helping on graphs: ‘natural selection favours cooperation if the benefit *B* of the act, divided by the cost *C*, exceeds the average number *k* of neighbours':

This rule has been derived using pair approximations for regular isothermal graphs and tested numerically as a lower bound for the evolution of helping on more complicated graphs.

Does inequality 1 provides us with a new pathway to the evolution of helping behaviours on graphs? We think that this is not the case. Indeed, kin selection operates whenever interactions occur among relatives, that is, among individuals that are more likely to inherit a strategy from a common ancestor than are individuals sampled at random from the population (Hamilton, 1964, 1970, 1971). This may happen when interactions take place among members of a family or when the population is structured through limited dispersal. In both cases, relatives remain close to each other. Because dispersal occurs only to the nearest neighbours in populations structured according to evolutionary graph theory, interactions occur necessarily among relatives.

In this paper, we carry out a retrospective analysis of the models for the evolution of helping on graphs of Ohtsuki *et al.* (2006) and Ohtsuki & Nowak (2006), and generalize them by applying inclusive fitness theory for finite populations (Rousset & Billiard, 2000; Rousset, 2004, 2006). This allows us to recover the mathematical results of evolutionary graph theory for ‘death–birth’ and ‘imitation’ life cycles as examples of inclusive fitness theory for spatially structured populations. In a companion paper, Grafen (2007) also provides an inclusive fitness analysis of the ‘death–birth’ and ‘birth–death’ life cycles of evolutionary graph theory. The reason for our retrospective analysis is to illustrate how results obtained heuristically by pair approximations can be obtained exactly by using inclusive fitness theory and to provide a link between the results of evolutionary graph theory and those of inclusive fitness theory. Further, by allowing more than one individual to live at each node of the graph and by allowing interactions to vary with the distance between nodes, our models allow us to represent different biological scenarios leading to the evolution of both helping and harming behaviours on graphs.