We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the "underlying treewidth" of a graph class $\mathcal{G}$ to be the minimum non-negative integer $c$ such that, for some function $f$, for every graph ${G \in \mathcal{G}}$ there is a graph $H$ with ${\text{tw}(H) \leq c}$ such that $G$ is isomorphic to a subgraph of ${H \boxtimes K_{f(\text{tw}(G))}}$. We introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth 3; the class of $K_{s,t}$-minor-free graphs has underlying treewidth $s$ (for ${t \geq \max\{s,3\}}$); and the class of $K_t$-minor-free graphs has underlying treewidth ${t-2}$. In general, we prove that a monotone class has bounded underlying treewidth if and only if it excludes some fixed topological minor. We also study the underlying treewidth of graph classes defined by an excluded subgraph or excluded induced subgraph. We show that the class of graphs with no $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a subdivided star, and that the class of graphs with no induced $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a star.
翻译:我们显示,许多带有约束树枝的图表可以被描述为以小树枝和一个约束大小完整图表绘制的强产图的子图。 如此, 定义一个图形类$\mathcal{G}$的“ 底植树枝” 是最小的非负整数 $c$。 对于每个图形 ${G\ in\mathcal{G ⁇ $, 对于某个函数, ${G\in\ in\mathcal{gr} $ 的图可以被描述为以美元为基底的硬产值 $(H)\leq c} 。 美元是树枝的硬值 $[H\boxwith] 。 我们引入了不连接的图层覆盖, 显示任何图表类的底基值。 使用此结果, 我们证明, 平面图的底底底值是$+美元 美元 底值的底值 。 如果我们直系的底值是底值 美元底值的底值, 那么平底值的平底值图表只有O值 。