Comparing Width Parameters on Graph Classes

TL;DR

The paper examines how the relative strength of six width parameters changes when restricting to graph classes such as $K_{t,t}$-subgraph-free graphs, line graphs, and their common superclass. It delivers complete equivalence results for several classes (notably $K_{t,t}$-subgraph-free graphs and line graphs) and a near-complete picture for $K_{t,t}$-free graphs, including a key theorem showing bounded mim-width implies bounded tree-independence number in $K_{t,t}$-free graphs. A central technical contribution is a tight link between line-graph width measures and the root graph’s treewidth, plus a lower bound connecting sim-width in line graphs to branch-width. The paper also analyzes how width parameters behave under graph powers, proving preservation of sim-width for odd powers and highlighting limits for even powers, with substantial algorithmic consequences such as XP algorithms and quadratic-time colorings on bounded-tree-independence-width classes. Overall, the results advance understanding of width-parameter hierarchies on sparse and dense graph classes and inform both structural theory and practical algorithm design.

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, 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.

Figures (3)

• Figure 1: Inclusion diagram of the three main graph classes considered in the paper, where $t \geq 3$.
• Figure 2: The relationships between the six different width parameters considered in the paper when restricted to different graph classes. A directed arrow from parameter $p$ to parameter $q$ indicates that $p$ dominates $q$, whereas a bidirected arrow indicates that $p$ and $q$ are equivalent. Although not explicitly stated in \ref{['t-main', 't-main2', 't-main3']}, all functions showing that a certain parameter $p$ dominates another parameter $q$ for a certain graph class $\mathcal{G}$ are computable and, as we will show in particular for the new results, can be obtained from the corresponding proofs.
• Figure 3: The graph $S_{2,3,4}+P_2+P_3+P_4$, which is an example of a graph that belongs to ${\cal S}$.

Theorems & Definitions (14)

• Theorem 1
• Corollary 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Corollary 8
• Corollary 9
• Corollary 10