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Graph width parameters: from structure to algorithm Shizhou Yang (1765555) PUREID: 617154195 width parameter mim-width sim-width hereditary graph class XP algorithm independent packings intersection graphs polynomial-time approximation schemes tree-independence number Solving a discrete optimization problem means seeking an optimal solution from finitely many options. Most discrete optimization problems are computationally hard. To overcome this, we may restrict the input and ask: Which input restrictions lead to "fast" algorithms? The input is often described by a graph and knowing that a graph is "easily decomposable" can be highly useful for designing efficient algorithms for many well-known optimization problems. A graph width parameter p is a function that assigns a number to each graph G, where a small value of p(G) usually means that the graph G is "easily decomposable" with respect to the parameter p. We use graph width parameters to show that certain graphs are "easily decomposable" with respect to specific graph width parameters and design "fast" algorithms for otherwise "hard" problems. In Chapter 2 and Chapter 3 we give basic definitions and background for the graph width parameters that are studied in this thesis, which are mim-width, sim-width, layered tree-independence number and the related notion of fractional tree-independence-number-fragility. In Chapter 4 we continue the work from Brettell et al. and prove (un)boundedness of mim-width for (H_1,H_2)-free graphs when H_1 is complete or edgeless, closing most of the open cases. In Chapter 5 we prove several other properties of sim-width. In Chapter 6 we study the relationships between sim-width, mim-width, tree-independence number, treewidth, clique-width and twin-width when restricted to K{t,t}-free graphs, K{t,t}-subgraph-free graphs and line graphs. In Chapter 7 we study the notion of fractional tree-independence-number-fragility and show that, for every graph in a fractionally fragile graph class, the meta-problem of finding a subset of its vertices satisfying a given CMSO2 formula and inducing a subgraph of bounded clique size admits a polynomial-time approximation scheme. 2026-10-01T16:31:37Z Text Thesis 10.17034/32640390.v1 https://figshare.com/articles/thesis/Graph_width_parameters_from_structure_to_algorithm/32640390 All Rights Reserved