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Abstract: Given a property (graph class) Π, a graph G, and an integer k, the Π-completion problem consists of deciding whether we can turn G into a graph with the property Π by adding at most k edges to G. The Π-completion problem is known to be NP-hard for general graphs when Π is the property of being a proper interval graph (PIG). In this work, we study the PIG-completion problem within different subclasses of chordal graphs. We show that the problem remains NP-complete even when restricted to split graphs. We then turn our attention to positive results and present polynomial time algorithms to solve the PIG-completion problem when the input is restricted to caterpillar and threshold graphs. We also present an efficient algorithm for the minimum co-bipartite-completion for quasi-threshold graphs, which provides a lower bound for the PIG-completion problem within this graph class.
Keywords: proper interval completion, split graph, threshold graph, quasi-threshold graph, caterpillar
Published in RUP: 06.08.2025; Views: 334; Downloads: 6
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