Abstract
For a graph property Π, i.e., a collection Π of graphs, the Connected Induced Π-Subgraph problem asks whether a graph G contains k vertices V′ such that the induced subgraph G[V′] is connected and belongs to Π.
In this paper, we regard k as a parameter and study the parameterized complexity of Connected Induced Π-Subgraph for hereditary properties Π. We give an almost complete characterization in terms of whether Π includes all complete graphs, all stars, or all paths: FPT if Π includes all complete graphs and stars, or excludes some complete graphs, stars and paths; and W[1]-hard otherwise (except the case that Π includes all complete graphs and paths but exclude some stars). For the remaining case, we show that it is W[1]-hard if Π includes all complete graphs K t , excludes a star K 1,s but includes all trees of maximum degree less than s. Our results imply a complete characterization for Π being H-free graphs for a fixed graph H: W[1]-hard if H is K t with t ≥ 3 or K 1,s with s ≥ 2, and FPT otherwise.
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Cai, L., Ye, J. (2014). Parameterized Complexity of Connected Induced Subgraph Problems. In: Gu, Q., Hell, P., Yang, B. (eds) Algorithmic Aspects in Information and Management. AAIM 2014. Lecture Notes in Computer Science, vol 8546. Springer, Cham. https://doi.org/10.1007/978-3-319-07956-1_20
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DOI: https://doi.org/10.1007/978-3-319-07956-1_20
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