Is there an operation on polyhedra that add $1$ vertex and $1$ facet (thus, due to Euler's formula, add two edges)?
Here, a polyhedron is the convex hull of a finite number of non coplanar points in ${\mathbb R^3}$. If $P$ has a face $F$ that is a $n$-gon, then the blow-up of that face (i.e. gluing a pyramid over the face) add $1$ vertex, but also add $n$ edges and $n-1$ facets. Since $n$ is at least $3$, it adds too much edges and facets for my purpose. Is there an operation that would fill my need?
Here is one idea: Look for a vertex $v$ that has exactly three adjacent vertices $a,b,c$, if one exists. Choose interior points points $a' \in \overline{va}$, $b' \in \overline{vb}$ and then cut your polytope $P$ with the plane that goes through $a',b,',c$. Then you will have replaced the single vertex $v$ with two vertices $a',b'$, and there will be one new facet, the triangle with vertices $a',b',c$. I'm not sure if this will do what you want, since I don't know what properties need to be preserved. Also, not every polytope has a vertex with exactly three neighboring vertices - but perhaps there is a way around this.