# On restricted completions of chordal and trivially perfect graphs

Let G be a graph having a vertex v such that H = G - v is a trivially perfect graph. We give a polynomial-time algorithm for the problem of deciding whether it is possible to add at most k edges to G to obtain a trivially perfect graph. This is a slight variation of the well-studied Edge Completion, also known as Minimum Fill-In, problem. We also show that if H is a chordal graph, then the problem of deciding whether it is possible to add at most k edges to G to obtain a chordal graph is -complete.

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