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We define a graph process $\mathcal{G}(p,q)$ based on a discrete branchingprocess with deletions and mergers, which is inspired by the 4-cycle structureof both the hypercube $Q_d$ and the lattice $\mathbb{Z}^d$ for large $d$.Individuals have Poisson offspring distribution with mean $1+p$ and certaindeletions and mergers occur with probability $q$; these parameters correspondto the mean number of edges discovered from a given vertex in an exploration ofa percolation cluster and to the probability that a non-backtracking path oflength four closes a cycle, respectively.  We prove survival and extinction under certain conditions on $p$ and $q$ thatheuristically match the known expansions of the critical probabilities for bondpercolation on the lattice $\mathbb{Z}^d$ and the hypercube $Q_d$. Theseexpansions have been rigorously established by Hara and Slade in 1995, and vander Hofstad and Slade in 2006, respectively. We stress that our method does notconstitute a branching process proof for the percolation threshold.  The analysis of the graph process survival is considerably more challengingthan for branching processes in discrete time, due to the interdependencebetween the descendants of different individuals in the same generation. Infact, it is left open whether the survival probability of $\mathcal{G}(p,q)$ ismonotone in $p$ or $q$; we discuss this and some other open problems regardingthe new graph process.

A branching process with deletions and mergers that matches the threshold for hypercube percolation