I am interested in the process of chordal completion that is the process of creating a chordal graph from a graph $G$ by being the smallest chordal graph that contains $G$ as a sub-graph.
I am in particular trying to find the maximal $k(n)$ such that for a family of graphs with $O(n)$ edges, the chordal completion requires $k(n)$ edges. I am trying to find the maximal $k(n)$, is this exponential in $n$? And is there an explicit construction?
What I have tried is that I think that I have examples where a graph with $n$ edges needs $2n$ edges for the chordal completion.