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 real 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：我们引入了随机图形$\mathcal{P}( n, q) $, 以两条长度为 $n\geq 1 美元路径的组合产生的随机图形$\ mathcal{P} (n, q) $, 其中一条路径的脊椎根据一个 Mallows 的折叠重新标签, 实际参数为 $0 < q(n)\leq 1$ 。 这个随机图形模型, 缠绕的路径, 经历一个进化过程: 如果 $ q 接近 $0 美元, 图形与一条路径相近, 且 q 美元 往往会变成一个扩张器 $ $ 。 为了了解 $\ macal{P} (n, q) 的进化过程, 我们确定$\ mathc{ p} (n, q) $(n, q) 最高到 log因子值的值的值。 我们还显示, 大小为 1 的分隔符的属性有一个直径阀值。 此外, 我们证明直线和垂直是 $q 具体值的直径的直径线号。