  We define a graph process $\mathcal{G}(p,q)$ based on a discrete branching
process with deletions and mergers, which is inspired by the 4-cycle structure
of both the hypercube $Q_d$ and the lattice $\mathbb{Z}^d$ for large $d$.
Individuals have Poisson offspring distribution with mean $1+p$ and certain
deletions and mergers occur with probability $q$; these parameters correspond
to the mean number of edges discovered from a given vertex in an exploration of
a percolation cluster and to the probability that a non-backtracking path of
length four closes a cycle, respectively.
  We prove survival and extinction under certain conditions on $p$ and $q$ that
heuristically match the known expansions of the critical probabilities for bond
percolation on the lattice $\mathbb{Z}^d$ and the hypercube $Q_d$. These
expansions have been rigorously established by Hara and Slade in 1995, and van
der Hofstad and Slade in 2006, respectively. We stress that our method does not
constitute a branching process proof for the percolation threshold.
  The analysis of the graph process survival is considerably more challenging
than for branching processes in discrete time, due to the interdependence
between the descendants of different individuals in the same generation. In
fact, it is left open whether the survival probability of $\mathcal{G}(p,q)$ is
monotone in $p$ or $q$; we discuss this and some other open problems regarding
the new graph process.
