Let G be a graph and k an integer with \(2\le k\le n\). The k -tree-connectivity of G, denoted by \(\kappa _k(G)\), is defined as the minimum \(\kappa _G(S)\) over all k-subsets S of vertices, where \(\kappa _G(S)\) denotes the maximum number of internally disjoint S-trees in G. The k -path-connectivity of G, denoted by \(\pi _k(G)\), is defined as the minimum \(\pi _G(S)\) over all k-subsets S of vertices, where \(\pi _G(S)\) denotes the maximum number of internally disjoint S-paths in G. In this paper, we first prove that for any fixed integer \(k\ge 1\), given a graph G and a subset S of V(G), deciding whether \(\pi _G(S)\ge k\) is \(\mathcal {NP}\)-complete. Moreover, we also show that for any fixed integer \(k_1\ge 5\), given a graph G, a \(k_1\)-subset S of V(G) and an integer \(1\le k_2\le n-1\), deciding whether \(\pi _G(S)\ge k_2\) is \(\mathcal {NP}\)-complete. Let \(\pi (k,\ell )=1+\max \{\kappa (G)|\ \)G\(\text {\ is\ a\ graph\ with}\ \pi _k(G)< \ell \}\). Hager (Discrete Math 59:53–59, 1986) showed that \(\ell (k-1)\le \pi (k,\ell )\le 2^{k-2}\ell\) and conjectured that \(\pi (k,\ell )=\ell (k-1)\) for \(k\ge 2\) and \(\ell \ge 1\). He also confirmed the conjecture for \(2\le k\le 4\) and proved \(\pi (5,\ell )\le \lceil \frac{9}{2}\ell \rceil\). By introducing a “Generalized Path-Bundle Transformation”, we confirm the conjecture for \(k=5\) and prove that \(\pi (k,\ell )\le 2^{k-3}\ell\) for \(k\ge 5\) and \(\ell \ge 1\). By employing this transformation, we also prove that if G is a graph with \(\kappa (G)\ge (k-1)\ell\) for any \(k\ge 2\) and \(\ell \ge 1\), then \(\kappa _k(G)\ge \ell\).

让G是一个图，k是一个整数，其中\(2\le k\le n\)。G的k 树连通性，用\(\kappa _k(G)\) 表示，定义为顶点的所有k 子集S上的最小值\(\kappa _G(S)\)，其中\(\ kappa _G(S)\)表示G中内部不相交的S树的最大数量。G的k路径连通性，表示为\(\pi _k(G)\)，定义为所有k 上的最小值\(\pi _G(S)\) 顶点的-subsets S，其中\(\pi _G(S)\)表示G中内部不相交的S -paths的最大数量。在本文中，我们首先证明，对于任何固定的整数\（K \ GE 1 \），给定图ģ和一个子集小号的V（G ^），决定是否\（\ PI _G（S）\锗的K \）是\(\mathcal {NP}\) -完成的。此外，我们还表明，对于任何固定的整数\（K_1 \ GE 5 \） ，给定图ģ，一个\（K_1 \） -subset小号的V（G ^) 和一个整数\(1\le k_2\le n-1\)，决定\(\pi _G(S)\ge k_2\)是否是\(\mathcal {NP}\) -完全的。设\(\pi (k,\ell )=1+\max \{\kappa (G)|\ \) G \(\text {\ is\ a\ graph\ with}\ \pi _k(G)< \ell \}\)。Hager (Discrete Math 59:53–59, 1986) 证明\(\ell (k-1)\le \pi (k,\ell )\le 2^{k-2}\ell\)并推测\ (\pi (k,\ell )=\ell (k-1)\)用于\(k\ge 2\)和\(\ell \ge 1\)。他还证实了\(2\le k\le 4\)的猜想并证明了\(\pi (5,\ell )\le \lceil \frac{9}{2}\ell \rceil\). 通过引入“广义路径束变换”，我们证实了\(k=5\)的猜想，并证明了\(\pi (k,\ell )\le 2^{k-3}\ell\)对于\(k\ge 5\)和\(\ell \ge 1\)。通过使用这种变换，我们还证明了如果G是一个具有\(\kappa (G)\ge (k-1)\ell\)对于任何\(k\ge 2\)和\(\ell \ge 1\)，然后\(\kappa _k(G)\ge \ell\)。