Let \(D=(V,A)\) be a digraph of order n, S a subset of V of size k and \(2\le k\le n\). A strong subgraph H of D is called an S-strong subgraph if \(S\subseteq V(H)\). A pair of S-strong subgraphs \(D_1\) and \(D_2\) are said to be arc-disjoint if \(A(D_1)\cap A(D_2)=\emptyset\). A pair of arc-disjoint S-strong subgraphs \(D_1\) and \(D_2\) are said to be internally disjoint if \(V(D_1)\cap V(D_2)=S\). Let \(\kappa _S(D)\) (resp. \(\lambda _S(D)\)) be the maximum number of internally disjoint (resp. arc-disjoint) S-strong subgraphs in D. The strong subgraph k -connectivity is defined as

As a natural counterpart of the strong subgraph k-connectivity, we introduce the concept of strong subgraph k -arc-connectivity which is defined as

A digraph \(D=(V, A)\) is called minimally strong subgraph \((k,\ell )\)-(arc-)connected if \(\kappa _k(D)\ge \ell\) (resp. \(\lambda _k(D)\ge \ell\)) but for any arc \(e\in A\), \(\kappa _k(D-e)\le \ell -1\) (resp. \(\lambda _k(D-e)\le \ell -1\)). In this paper, we first give complexity results for \(\lambda _k(D)\), then obtain some sharp bounds for the parameters \(\kappa _k(D)\) and \(\lambda _k(D)\). Finally, minimally strong subgraph \((k,\ell )\)-connected digraphs and minimally strong subgraph \((k,\ell )\)-arc-connected digraphs are studied.

令\（D =（V，A）\）是n阶的有向图，S是大小为k的V的子集和\（2 \ le k \ le n \）。甲强子图ħ的d被称为小号-强子图，如果\（S \ subseteq V（1H）\） 。如果\（A（D_1）\ cap A（D_2）= \ emptyset \），则一对S -strong子图\（D_1 \）和\（D_2 \）被称为弧不相交的。一对弧形不相交的S-强子图\（D_1 \）和\（D_2 \）如果\（V（D_1）\ cap V（D_2）= S \）被称为内部不相交。让\（\卡帕_S（d）\） （相应地，\（\拉姆达_S（d）\） ）在内部不相交的最大数目（分别为圆弧不相交）š在-strong子图d。所述强子图ķ -connectivity被定义为

作为强子图k-连通性的自然对应物，我们介绍了强子图 k -arc-连通性的概念，其定义为

有向图\（D =（V，A）\）被称为最小强子图 \（（k，\ ell）\）-（arc-）如果\（\ kappa _k（D）\ ge \ ell \）（分别为\（\ lambda _k（D）\ ge \ ell \）），但对于任何弧线\（e \ in A \），\（\ kappa _k（De）\ le \ ell -1 \）（resp。\ （\ lambda _k（De）\ le \ ell -1 \））。在本文中，我们首先给出\（\ lambda _k（D）\）的复杂度结果，然后为参数\（\ kappa _k（D）\）和\（\ lambda _k（D）\）获得一些尖锐边界。最后，最小强子图\（（k，\ ell）\）相连的有向图和最小强子图\（（k，\ ell）\）-连接弧的有向图。