We introduce a new class of interacting particle systems on a graph G. Suppose initially there are N
i
(0) particles at each vertex i of G, and that the particles interact to form a Markov chain: at each instant two particles are chosen at random, and if these are at adjacent vertices of G, one particle jumps to the other particle's vertex, each with probability 1/2. The process
N
enters a death state after a finite time when all the particles are in some independent subset of the vertices of G, i.e. a set of vertices with no edges between any two of them. The problem is to find the distribution of the death state, η
i
= N
i
(∞), as a function of N
i
(0).
We are able to obtain, for some special graphs, the limiting distribution of N
i
if the total number of particles N → ∞ in such a way that the fraction, N
i
(0)/S = ξ
i
, at each vertex is held fixed as N → ∞. In particular we can obtain the limit law for the graph S
2, the two-leaf star which has three vertices and two edges.