同步图参数及其界的研究
Simultaneous Graph Parameters and How to Bound Them
AI总结：
该研究探讨同步𝓒-数与其他图参数的关系，明确了具备和不具备特定有界性性质的图参数，刻画了可作为同步𝓒-数上界的参数及对应图类，还给出相关算法结果。
AI中文摘要：
Beisegel等人[SWAT 2024]提出了同步𝓒-数的概念，该概念将图类𝓒与一个图参数关联起来。对于给定图G，同步𝓒-数是满足以下条件的最小数d：存在图H∈𝓒，以及函数L: V(G)→𝒫({1,…,d})，使得G中顶点u和v相邻当且仅当它们在H中相邻且集合L(u)与L(v)不相交。我们研究这些同步𝓒-数与其他图参数的关系，特别探究哪些参数满足如下性质：参数p在图类𝓒上有界，当且仅当对任意固定的d，p在同步𝓒-数为d的图类上有界。我们证明了许多知名图参数具有该性质，例如团宽（cliquewidth）、孪生宽（twin-width）、mim宽（mim-width）、树独立数、薄度（thinness）以及盒维数（boxicity）；同时也给出了不具备该性质的参数，包括模块宽（modular-width）和树长（tree-length）。我们还研究了参数何时构成同步𝓒-数的上界，刻画了图类𝓒使得树宽（treewidth）、路宽（pathwidth）、带宽（bandwidth）和树深度（treedepth）这几个参数可作为同步𝓒-数的上界；此外，给出了图类𝓒的充分条件，使得当𝓟替换为完全图、无边图、余图（cographs）或图类𝓒本身时，𝓟-模块基数（𝓟-modular cardinality）可作为同步𝓒-数的上界。相反，我们证明模块宽永远无法作为非平凡同步𝓒-数的上界。最后，我们给出了关于团问题和同步𝓒-数计算的若干通用算法结果。
英文摘要：
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.
