We prove that for every $t \in \mathbb{N}$, the graph $K_{2,t}$ satisfies the fat minor conjecture of Georgakopoulos and Papasoglu: for every $K\in \mathbb{N}$ there exist $M,A\in \mathbb{N}$ such that every graph with no $K$-fat $K_{2,t}$ minor is $(M,A)$-quasi-isometric to a graph with no $K_{2,t}$ minor. We use this to obtain an efficient algorithm for approximating the minimal multiplicative distortion of any embedding of a finite graph into a $K_{2,t}$-minor-free graph, answering a question of Chepoi, Dragan, Newman, Rabinovich, and Vax\`es from 2012.

翻译：我们证明对于任意$t \in \mathbb{N}$，图$K_{2,t}$满足Georgakopoulos与Papasoglu提出的稠密子式猜想：对于任意$K\in \mathbb{N}$，存在$M,A\in \mathbb{N}$，使得每个不包含$K$-稠密$K_{2,t}$子式的图都与某个不包含$K_{2,t}$子式的图是$(M,A)$-拟等距的。利用这一结论，我们提出了一种高效算法，用于近似计算任意有限图嵌入到不含$K_{2,t}$子式的图中的最小乘法失真度，从而解决了Chepoi、Dragan、Newman、Rabinovich和Vaxès在2012年提出的问题。