We investigate a relaxation of the notion of treewidth-fragility, namely tree-independence-number-fragility. In particular, we obtain polynomial-time approximation schemes for independent packing problems on fractionally tree-independence-number-fragile graph classes. Our approach unifies and extends several known polynomial-time approximation schemes on seemingly unrelated graph classes, such as classes of intersection graphs of fat objects in a fixed dimension or proper minor-closed classes. We also study the related notion of layered tree-independence number, a relaxation of layered treewidth.

Mim-width and sim-width are among the most powerful graph width parameters, with sim-width more powerful than mim-width, which is in turn more powerful than clique-width. While several NP-hard graph problems become tractable for graph classes whose mim-width is bounded and quickly computable, no algorithmic applications of boundedness of sim-width are known. In Kang et al. (2017) [32], it is asked whether Independent Set and 3-Colouring are NP-complete on graphs of sim-width at most 1. We observe that, for each k∈N, List k-Colouring is polynomial-time solvable for graph classes whose sim-width is bounded and quickly computable. Moreover, we show that if the same holds for Independent Set, then Independent H-Packing is polynomial-time solvable for graph classes whose sim-width is bounded and quickly computable. This problem is a common generalisation of Independent Set, Induced Matching, Dissociation Set and k-Separator. We also make progress toward classifying the mim-width of (H1,H2)-free graphs in the case H1 is complete or edgeless. Our results solve some open problems in Brettell et al. (2022) [6].