We introduce the random graph $\mathcal{P}(n,q)$ which results from taking the union of two paths of length $n\geq 1$, where the vertices of one of the paths have been relabelled according to a Mallows permutation with parameter $0<q(n)\leq 1$. This random graph model, the tangled path, goes through an evolution: if $q$ is close to $0$ the graph bears resemblance to a path, and as $q$ tends to $1$ it becomes an expander. In an effort to understand the evolution of $\mathcal{P}(n,q)$ we determine the treewidth and cutwidth of $\mathcal{P}(n,q)$ up to log factors for all $q$. We also show that the property of having a separator of size one has a sharp threshold. In addition, we prove bounds on the diameter, and vertex isoperimetric number for specific values of $q$.

翻译：我们引入了随机图$\mathcal{P}(n,q)$，该图由两条长度为$n\geq 1$的路径取并集得到，其中一条路径的顶点根据参数$0<q(n)\leq 1$的Mallows排列重新标注。这一随机图模型（交织路径）经历了一个演化过程：当$q$接近0时，图与路径相似；而当$q$趋近1时，图变为扩张器。为理解$\mathcal{P}(n,q)$的演化，我们确定了对于所有$q$，$\mathcal{P}(n,q)$的树宽和割宽（在对数因子范围内）。我们还证明了具有大小为1的分隔符这一性质存在尖锐阈值。此外，针对特定$q$值，我们给出了直径和顶点等周数的界限。