Comparing width parameters on graph classes

We study how the relationship between non-equivalent width parameters changes once we restrict to some special graph class. As width parameters we consider treewidth, clique-width, twin-width, mim-width, sim-width and tree-independence number, whereas as graph classes we consider $K_{t,t}$-subgraph-free graphs, line graphs and their common superclass, for $t\ge 3$, of $K_{t,t}$-free graphs. For $K_{t,t}$-subgraph-free graphs, we extend a known result of Gurski and Wanke (2000) and provide a complete comparison, showing in particular that treewidth, clique-width, mim-width, sim-width and tree-independence number are all equivalent. For line graphs, we extend a result of Gurski and Wanke (2007) and also provide a complete comparison, showing in particular that clique-width, mim-width, sim-width and tree-independence number are all equivalent, and bounded if and only if the class of root graphs has bounded treewidth. For $K_{t,t}$-free graphs, we provide an almost-complete comparison, leaving open only one missing case. We show in particular that $K_{t,t}$-free graphs of bounded mim-width have bounded tree-independence number, and obtain structural and algorithmic consequences of this result, such as a proof of a special case of a recent conjecture of Dallard, Milanič and Štorgel. Finally, we consider the question of whether boundedness of a certain width parameter is preserved under graph powers. We show that this question has a positive answer for sim-width precisely in the case of odd powers.