Algorithms for subgraph complementation to some classes of graphs
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For a class 𝒢 of graphs, the objective of Subgraph Complementation to 𝒢 is to find whether there exists a subset S of vertices of the input graph G such that modifying G by complementing the subgraph induced by S results in a graph in 𝒢. We obtain a polynomial-time algorithm for the problem when 𝒢 is the class of graphs with minimum degree at least k, for a constant k, answering an open problem by Fomin et al. (Algorithmica, 2020). When 𝒢 is the class of graphs without any induced copies of the star graph on t+1 vertices (for any constant t≥ 3) and diamond, we obtain a polynomial-time algorithm for the problem. This is in contrast with a result by Antony et al. (Algorithmica, 2022) that the problem is NP-complete and cannot be solved in subexponential-time (assuming the Exponential Time Hypothesis) when 𝒢 is the class of graphs without any induced copies of the star graph on t+1 vertices, for every constant t≥ 5.
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