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 $t \geq 3$, of $K_{t,t}$-free graphs. We first provide a complete comparison when restricted to $K_{t,t}$-subgraph-free graphs, showing in particular that treewidth, clique-width, mim-width, sim-width and tree-independence number are all equivalent. This extends a result of Gurski and Wanke (2000) stating that treewidth and clique-width are equivalent for the class of $K_{t,t}$-subgraph-free graphs. Next, we provide a complete comparison when restricted to line graphs, showing in particular that, on any class of line graphs, 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. This extends a result of Gurski and Wanke (2007) stating that a class of graphs ${\cal G}$ has bounded treewidth if and only if the class of line graphs of graphs in ${\cal G}$ has bounded clique-width. We then provide an almost-complete comparison for $K_{t,t}$-free graphs, leaving one missing case. Our main result is that $K_{t,t}$-free graphs of bounded mim-width have bounded tree-independence number. This result has structural and algorithmic consequences. In particular, it proves a special case of a conjecture of Dallard, Milani\v{c} and \v{S}torgel. Finally, we consider the question of whether boundedness of a certain width parameter is preserved under graph powers. We show that the question has a positive answer for sim-width precisely in the case of odd powers.

翻译：我们研究了当限制在特定图类时，非等价宽度参数之间关系的变化。作为宽度参数，我们考虑了树宽、团宽、孪生宽、MIM宽、SIM宽和树独立数；作为图类，我们考虑了$K_{t,t}$-子图自由图、线图及其共同超类（对于$t \geq 3$的$K_{t,t}$-自由图）。首先，我们在限制于$K_{t,t}$-子图自由图时给出了完整的比较，特别表明树宽、团宽、MIM宽、SIM宽和树独立数均等价。这推广了Gurski与Wanke（2000）的一个结果，即树宽与团宽在$K_{t,t}$-子图自由图类上等价。其次，我们在限制于线图时给出了完整的比较，特别表明在任何线图类上，团宽、MIM宽、SIM宽和树独立数均等价，且有界当且仅当根图类具有有界树宽。这推广了Gurski与Wanke（2007）的一个结果，即图类$\mathcal{G}$具有有界树宽当且仅当$\mathcal{G}$中图的线图类具有有界团宽。接着，我们对$K_{t,t}$-自由图给出了近乎完整的比较，仅遗留了一个未解决情形。我们的主要结果是：具有有界MIM宽的$K_{t,t}$-自由图具有有界树独立数。这一结果具有结构性和算法性推论，特别地，它证明了Dallard、Milani\v{c}与\v{S}torgel猜想的一个特例。最后，我们考虑了在某些宽度参数的有界性是否在图幂运算下保持的问题，并表明该问题对于SIM宽仅在奇次幂情形下有肯定答案。