In the last decade, algorithmic frameworks based on a structural graph parameter called mim-width have been developed to solve generally NP-hard problems. However, it is known that the frameworks cannot be applied to the Clique problem, and the complexity status of many problems of finding dense induced subgraphs remains open when parameterized by mim-width. In this paper, we investigate the complexity of the problem of finding a maximum induced subgraph that satisfies prescribed properties from a given graph with small mim-width. We first give a meta-theorem implying that various induced subgraph problems are NP-hard for bounded mim-width graphs. Moreover, we show that some problems, including Clique and Induced Cluster Subgraph, remain NP-hard even for graphs with (linear) mim-width at most 2. In contrast to the intractability, we provide an algorithm that, given a graph and its branch decomposition with mim-width at most 1, solves Induced Cluster Subgraph in polynomial time. We emphasize that our algorithmic technique is applicable to other problems such as Induced Polar Subgraph and Induced Split Subgraph. Since a branch decomposition with mim-width at most 1 can be constructed in polynomial time for block graphs, interval graphs, permutation graphs, cographs, distance-hereditary graphs, convex graphs, and their complement graphs, our positive results reveal the polynomial-time solvability of various problems for these graph classes.

翻译：过去十年中，基于称为mim宽度的结构图参数开发的算法框架被用于解决通常为NP难的问题。然而，已知这些框架无法应用于团问题，且许多寻找稠密诱导子图问题的复杂度状态在通过mim宽度参数化时仍未解决。本文研究了在给定具有小mim宽度的图中寻找满足特定性质的最大诱导子图问题的复杂度。我们首先给出一个元定理，表明多种诱导子图问题对于有界mim宽度图是NP难的。此外，我们证明包括团问题和诱导聚类子图在内的一些问题，即使对于mim宽度至多为2（线性）的图仍然是NP难的。与这些难解性结果相对，我们提出一种算法，当给定图及其mim宽度至多为1的分支分解时，能在多项式时间内求解诱导聚类子图问题。我们强调该算法技术可推广至其他问题，如诱导极子图和诱导分裂子图问题。由于对于块图、区间图、置换图、补图、距离遗传图、凸图及其补图，可以在多项式时间内构造出mim宽度至多为1的分支分解，我们的积极结果揭示了这些图类上多种问题的多项式时间可解性。