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- W4377371801 abstract "It is currently an unsolved problem to determine whether a $triangle$-free planar graph $G$ contains an independent set $A$ such that $G[V_Gsetminus A]$ is $2$-choosable. However, in this paper, we take a slightly different approach by relaxing the planarity condition. We prove the $mathbb{NP}$-completeness of the above decision problem when the graph is $triangle$-free, $4$-colorable, and of diameter $3$. Building upon this notion, we examine the computational complexity of two optimization problems: minimum near $3$-choosability and minimum $2$-choosable deletion. In the former problem, the goal is to find an independent set $A$ of minimum size in a given graph $G$, such that the induced subgraph $G[V_G setminus A]$ is $2$-choosable. We establish that this problem is $mathbb{NP}$-hard to approximate within a factor of $|V_G|^{1-epsilon}$ for any $epsilon > 0$, even for planar bipartite graphs. On the other hand, the problem of minimum $2$-choosable deletion involves determining a vertex set $A subseteq V_G$ of minimum cardinality such that the induced subgraph $G[V_G setminus A]$ is $2$-choosable. We prove that this problem is $mathbb{NP}$-complete, but can be approximated within a factor of $O(log |V_G|)$." @default.
- W4377371801 created "2023-05-23" @default.
- W4377371801 creator A5031083769 @default.
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- W4377371801 date "2023-05-19" @default.
- W4377371801 modified "2023-09-30" @default.
- W4377371801 title "Complexity of Near-3-Choosability Problem" @default.
- W4377371801 doi "https://doi.org/10.48550/arxiv.2305.11607" @default.
- W4377371801 hasPublicationYear "2023" @default.
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