  We study a discrete dynamic on weighted bipartite graphs on a torus,
analogous to dimer integrable systems in Goncharov-Kenyon 2013. The dynamic on
the graph is an urban renewal together with shrinking all 2-valent vertices,
while it is a cluster transformation on the weight. The graph is not necessary
obtained from an integral polygon. We show that all Hamiltonians, partition
functions of all weighted perfect matchings with a common homology class, are
invariant under a move on the weighted graph. This move coincides with a
cluster mutation, analog to Y-seed mutation in dimer integrable systems. We
construct graphs for Q-systems of type A and B and show that the Hamiltonians
are conserved quantities of the systems. The conserved quantities can be
written as partition functions of hard particles on a certain graph. For type
A, they Poisson commute under a nondegenerate Poisson bracket.
