Comparing Width Parameters on Graph Classes

2023-08-10

arXiv.org

AI Summary

This study systematically compares graph width parameters—including treewidth, clique-width, twin-width, mim-width, sim-width, and tree-independence number—on $K_{t,t}$-subgraph-free graphs, line graphs, and their superclass of $K_{t,t}$-free graphs. Using structural graph theory, forbidden-subgraph characterizations, and analysis of powers of line graphs, we establish several novel results: (i) In $K_{t,t}$-free graphs, bounded mim-width implies bounded tree-independence number, partially resolving the Dallard–Milanič–Štorgel conjecture; (ii) On line graphs, multiple width parameters are shown to be equivalent and tightly coupled with the treewidth of the root graph; (iii) A complete comparative hierarchy of all major width parameters is established for $K_{t,t}$-subgraph-free graphs, and an almost complete comparison is achieved for $K_{t,t}$-free graphs; (iv) Sim-width is proven to remain bounded under odd powers of graphs.

Abstract

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, Milaniv{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.

Problem

• Compare width parameters on K_{t,t}-subgraph-free graphs
• Analyze width parameter equivalence in line graphs
• Investigate boundedness preservation under graph powers

Innovation

• Compares width parameters on Kt,t-subgraph-free graphs
• Equates clique-width, mim-width, sim-width for line graphs
• Shows bounded mim-width implies bounded tree-independence number