The Bondy-Chvatal closure is independent of the order of eligible edge additions

If $G$ has no nonedges, no addition is possible and every maximal sequence terminates at $G$.

Assume the claimed uniqueness holds for every graph with fewer than $k$ nonedges, and let $G$ have $k>0$ nonedges. Once a nonedge is eligible, it remains eligible after other edges are added because degrees do not decrease.

If $G$ has no eligible nonedge, every maximal sequence is empty. Otherwise compare two maximal sequences, whose first added edges are $e$ and $f$. If $e=f$, their tails start at the same graph $G+e$, which has fewer than $k$ nonedges, so the induction hypothesis gives the same terminal graph.

If $e\mathrm{\ne }f$, persistence from step 1.2 makes $f$ eligible after adding $e$ and $e$ eligible after adding $f$. Thus both first-step graphs can legally reach $G+e+f$. Applying the induction hypothesis inside $G+e$ compares its given tail with the tail beginning by $f$; applying it inside $G+f$ compares its given tail with the tail beginning by $e$. Both therefore have the same terminal graph as a maximal continuation from $G+e+f$.

The cases in steps 2.1 and 2.2 establish uniqueness for graphs with $k$ nonedges. By induction, all finite graphs have a unique terminal graph, so the closure is independent of the addition order.