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Given a property (graph class) $\Pi$, a graph $G$, and an integer $k$, the\emph{$\Pi$-completion} problem consists in deciding whether we can turn $G$into a graph with the property $\Pi$ by adding at most $k$ edges to $G$. The$\Pi$-completion problem is known to be NP-hard for general graphs when $\Pi$is the property of being a proper interval graph (PIG). In this work, we studythe PIG-completion problem %when $\Pi$ is the class of proper interval graphs(PIG) within different subclasses of chordal graphs. We show that the problemremains NP-complete even when restricted to split graphs. We then turn ourattention to positive results and present polynomial time algorithms to solvethe PIG-completion problem when the input is restricted to caterpillar andthreshold graphs. We also present an efficient algorithm for the minimumco-bipartite-completion for quasi-threshold graphs, which provides a lowerbound for the PIG-completion problem within this graph class.

On the proper interval completion problem within some chordal subclasses