For a graph family~${\cal F}$, the minimum ${\cal F}$-completion problem asks for the minimum number of non-edges to be added to a given graph~$G$ so that the resulting graph belongs to the family~${\cal F}$. This problem is NP-complete for many subclasses of perfect graphs, while polynomial solutions are available for minimal completion sets or restricted input. We study the minimum $P_4$-sparse completion problem that arises after adding a prescribed non-edge to a \(P_4\)-sparse graph \(G\), with the resulting graph required to remain \(P_4\)-sparse. For the case in which $G$ consists of two connected components, each containing one endpoint of the added non-edge, we characterize the possible forms of an optimal solution. Combining this characterization with the solution of the other cases identified by the structural decomposition of \(P_4\)-sparse graphs, we obtain an algorithm for the general problem that runs in $O(n^2)$ time and $O(n)$ space.

翻译：暂无翻译