Comparing Width Parameters on Graph Classes
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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 ≥ 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 G has bounded treewidth if and only if the class of line graphs of graphs in 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, Milanič and Štorgel. 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.
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