Complexity of Near-3-Choosability Problem

Abstract

It is currently an unsolved problem to determine whether, for every 2-list assignment L of a $\triangle$-free planar graph G, there exists an independent set $A_L$ such that $G[V_G{\setminus } A_L]$ is L-colorable. However, in this paper, we take a slightly different approach to the above problem. We prove the ${\mathbb {N}}{\mathbb {P}}$-completeness of the decision problem of determining an independent set A such that $G[V_G{\setminus } A]$ is 2-choosable for $\triangle$-free, 4-colorable graphs of diameter 3. Building upon this notion, we examine the computational complexity of two optimization problems: minimum near-3-choosability and minimum 2-choosable-edge-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 ${\mathbb {N}}{\mathbb {P}}$-hard to approximate within a factor of $|V_G|^{1-\epsilon }$ for any $\epsilon > 0$, for planar bipartite graphs of arbitrary large girth. On the other hand, the problem of minimum 2-choosable-edge-deletion involves determining an edge set $F \subseteq E_G$ of minimum cardinality such that the spanning subgraph $G[E_G {\setminus } F]$ is 2-choosable. We prove that this problem can be approximated within a factor of $O(\log |V_G|)$.