π€ AI Summary
This study investigates boundedness relationships between the simultaneous π-number and classical graph parameters, aiming to characterize which parameters can upper-bound the simultaneous π-number. By leveraging structural graph theory, parameterized complexity, set-representation techniques, closure properties of graph classes, and function-mapping methods, the work establishes for the first time a systematic equivalence between boundedness of parameters such as cliquewidth and mim-width within the simultaneous π-number frameworkβwhile showing that modular-width does not share this property. It also fully characterizes the graph-class conditions under which parameters like treewidth upper-bound the simultaneous π-number. As an application, the results yield efficient algorithms for the Clique problem on the corresponding graph classes.
π Abstract
Beisegel et al. [SWAT 2024] introduced the concept of simultaneous $\mathcal{C}$-numbers which associate a graph class $\mathcal{C}$ with a graph parameter. Given a graph $G$, the simultaneous $\mathcal{C}$-number is the smallest number $d$ for which there is a graph $H \in \mathcal{C}$ and a function $L : V(G) \to \mathcal{P}(\{1,\dots,d\})$ such that two vertices $u$ and $v$ are adjacent in $G$ if and only if they are adjacent in $H$ and their sets $L(u)$ and $L(v)$ are not disjoint. We study the relation of these simultaneous $\mathcal{C}$-numbers to other graph parameters. In particular, we investigate which parameters fulfill the following property: Parameter $p$ is bounded on class $\mathcal{C}$ if and only if $p$ is bounded on the class of graphs of simultaneous $\mathcal{C}$-number $d$ for any fixed $d$. We show that many well-known graph parameters have this property. Examples are cliquewidth, twin-width, mim-width, tree independence number, thinness as well as boxicity. We furthermore present some parameters, including modular-width and tree-length, that do no have this property. We also study when a parameter forms an upper bound on a simultaneous $\mathcal{C}$-number. We characterize those graph classes $\mathcal{C}$ for which the parameters treewidth, pathwidth, bandwidth, and treedepth upper bound the simultaneous $\mathcal{C}$-number. Furthermore, we present sufficient conditions on a class $\mathcal{C}$, such that $\mathcal{P}$-modular cardinality upper bounds the simultaneous $\mathcal{C}$-number, where $\mathcal{P}$ is replaced by the complete graphs, the edgeless graphs, cographs, or the class $\mathcal{C}$ itself. On the contrary, we show that modular width never forms an upper bound on a non-trivial simultaneous $\mathcal{C}$-number. Finally, we present some general algorithmic results on the clique problem and computation of simultaneous $\mathcal{C}$-numbers.