What is the complexity of chordalization? A graph $G=(V,E)$ is a chordal graph, if it does not contain an induced
cycle of length at least four. We say a graph $H$ is a chordalization of graph
$G$, if $H$ contains $G$ as a subgraph, and $H$ is chordal. 

$Q_1$: Find minimum number of edges whose addition to a given graph makes the graph a chordal graph.

According to this, $Q_1$ is NP-hard.

$Q_2$: Find a chordalization that does not introduce new $K_4$? 

What is the complexity of $Q_2$? Is $Q_2$ harder than $Q_1$?

{
Remark: After Florent comment, I changed $Q_1$ from the following: 

$Q_1$ in first version of my post: What is the complexity of giving an arbitrary chordalization of input graph?
}
If the graph doesn't already have any K4, then Q2 is the same as asking for whether the graph has treewidth at most two. This is easy to check in linear time: it's true iff every biconnected component is series-parallel.