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 resut 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的关系如何变化。作为宽度参数，我们考虑树宽、团宽、双胞胎宽、mim-宽、sim-宽和树独立数；作为图类，我们考虑$K_{t,t}$-子图自由图、线图及其公共超类（即$t \geq 3$时的$K_{t,t}$-自由图）。首先，我们提供了限制在$K_{t,t}$-子图自由图上的完整比较，特别表明树宽、团宽、mim-宽、sim-宽和树独立数在此类上均等价。这推广了Gurski和Wanke（2000）关于树宽与团宽在$K_{t,t}$-子图自由图类上等价的结论。接着，我们提供了限制在线图上的完整比较，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，并且有界当且仅当根图类具有有界树宽。这推广了Gurski和Wanke（2007）关于图类${\cal G}$的树宽有界当且仅当${\cal G}$中线图的团宽有界的结论。随后，我们对$K_{t,t}$-自由图给出了几乎完整的比较，仅留下一个未决情形。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。该结果具有结构和算法上的意义，特别证明了Dallard、Milani\v{c}和\v{S}torgel猜想的一个特例。最后，我们探讨了某个宽度参数的有界性在图的幂运算下是否保持不变的问题。我们证明，对于sim-宽，该问题仅在奇次幂情形下具有肯定答案。