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- W4309783453 abstract "Abstract Covering a graph with cohesive subgraphs is a classical problem in theoretical computer science, for example when the cohesive subgraph model considered is a clique. In this paper, we consider as a model of cohesive subgraph the 2-clubs, which are induced subgraphs of diameter at most 2. We prove new complexity results on the $$mathsf {Min~2text {-}Club~Cover}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>Min</mml:mi> <mml:mspace /> <mml:mn>2</mml:mn> <mml:mtext>-</mml:mtext> <mml:mi>Club</mml:mi> <mml:mspace /> <mml:mi>Cover</mml:mi> </mml:mrow> </mml:math> problem, a variant recently introduced in the literature which asks to cover the vertices of a graph with a minimum number of 2-clubs. First, we answer an open question on the decision version of $$mathsf {Min~2text {-}Club~Cover}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>Min</mml:mi> <mml:mspace /> <mml:mn>2</mml:mn> <mml:mtext>-</mml:mtext> <mml:mi>Club</mml:mi> <mml:mspace /> <mml:mi>Cover</mml:mi> </mml:mrow> </mml:math> that asks if it is possible to cover a graph with at most two 2-clubs, and we prove that it is W[1]-hard when parameterized by the distance to a 2-club. Then, we consider the complexity of $$mathsf {Min~2text {-}Club~Cover}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>Min</mml:mi> <mml:mspace /> <mml:mn>2</mml:mn> <mml:mtext>-</mml:mtext> <mml:mi>Club</mml:mi> <mml:mspace /> <mml:mi>Cover</mml:mi> </mml:mrow> </mml:math> on some graph classes. We prove that $$mathsf {Min~2text {-}Club~Cover}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>Min</mml:mi> <mml:mspace /> <mml:mn>2</mml:mn> <mml:mtext>-</mml:mtext> <mml:mi>Club</mml:mi> <mml:mspace /> <mml:mi>Cover</mml:mi> </mml:mrow> </mml:math> remains NP-hard on subcubic planar graphs, W[2]-hard on bipartite graphs when parameterized by the number of 2-clubs in a solution, and fixed-parameter tractable on graphs having bounded treewidth." @default.
- W4309783453 created "2022-11-29" @default.
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- W4309783453 date "2022-11-21" @default.
- W4309783453 modified "2023-09-25" @default.
- W4309783453 title "On the Tractability of Covering a Graph with 2-Clubs" @default.
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