One of the most fundamental results in structural graph theory is the “two-paths theorem” that characterizes 2-linkage by planarity. As an extension of the theorem, we consider the following problem for a fixed graph H with four vertices: Given a graph G and an injective map from $V(H)$ to $V(G)$, is there a subdivision of H in G with four branch vertices specified by the map? Hence the case $H=2K_2$ corresponds to the 2-linkage problem. In this paper, for any fixed H with four vertices, we give a structural characterization of 6-connected graphs G with no such subdivision of H. As a corollary, we prove that every 7-connected graph contains a subdivision of $K_4$ with prescribed branch vertices. This generalizes a result of McCarty, Wang and Yu which states that every 7-connected graph is 4-ordered. We also prove that every triangle-free 6-connected graph contains a subdivision of $K_4$ with prescribed branch vertices. This solves a special case of a conjecture of Mader.

结构图论中最基本的结果之一是“双路径定理”，它通过平面性来表征 2-链接。作为定理的一个扩展，我们考虑一个有四个顶点的固定图H的以下问题： 给定一个图G和一个来自$五(H)$ 到 $五(G)$，有一个细分ħ在ģ与由地图指定了四个分支的顶点？因此情况$H=2{ķ}_2$对应于 2-linkage 问题。在本文中，对于任何具有四个顶点的固定H，我们给出了没有H细分的6 连通图G的结构表征。作为推论，我们证明了每个 7 连通图都包含${ķ}_4$具有规定的分支顶点。这概括了 McCarty、Wang 和 Yu 的一个结果，即每个 7 连通图都是 4 阶的。我们还证明了每个无三角形的 6 连通图都包含${ķ}_4$具有规定的分支顶点。这解决了马德猜想的一个特例。