Abstract
Let d and k be two given integers, and let G be a graph. Can we reduce the independence number of G by at least d via at most k graph operations from some fixed set S? This problem belongs to a class of so-called blocker problems. It is known to be co-NP-hard even if S consists of either an edge contraction or a vertex deletion. We further investigate its computational complexity under these two settings:
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we give a sufficient condition on a graph class for the vertex deletion variant to be co-NP-hard even if \(d=k=1\);
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in addition we prove that the vertex deletion variant is co-NP-hard for triangle-free graphs even if \(d=k=1\);
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we prove that the edge contraction variant is NP-hard for bipartite graphs but linear-time solvable for trees.
By combining our new results with known ones we are able to give full complexity classifications for both variants restricted to H-free graphs.
D. Paulusma received support from EPSRC (EP/K025090/1).
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Paulusma, D., Picouleau, C., Ries, B. (2017). Blocking Independent Sets for H-Free Graphs via Edge Contractions and Vertex Deletions. In: Gopal, T., Jäger , G., Steila, S. (eds) Theory and Applications of Models of Computation. TAMC 2017. Lecture Notes in Computer Science(), vol 10185. Springer, Cham. https://doi.org/10.1007/978-3-319-55911-7_34
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