Strong Subgraph Connectivity of Digraphs

Abstract Let $$D=(V,A)$$ D = ( V , A ) be a digraph of order n , S subset V size k and $$2\le k\le n$$ 2 ≤ k n . A strong subgraph H D is called an - if $$S\subseteq V(H)$$ S ⊆ H pair -strong subgraphs $$D_1$$ 1 $$D_2$$ are said to arc-disjoint $$A(D_1)\cap A(D_2)=\emptyset$$ ∩ ∅ internally disjoint $$V(D_1)\cap V(D_2)=S$$ $$\kappa _S(D)$$ κ (resp. $$\lambda λ ) the maximum number arc-disjoint) in The -connectivity defined as $$\begin{aligned} \kappa _k(D)=\min \{\kappa _S(D)\mid S\subseteq V, |S|=k\}. \end{aligned}$$ min { ∣ | } . As natural counterpart -connectivity, we introduce concept -arc-connectivity which \lambda \{\lambda V(D), $$D=(V, A)$$ minimally $$(k,\ell )$$ ℓ -(arc-)connected _k(D)\ge \ell$$ ≥ but for any arc $$e\in A$$ e ∈ _k(D-e)\le \ell -1$$ - ). In this paper, first give complexity results _k(D)$$ then obtain some sharp bounds parameters Finally, -connected digraphs -arc-connected studied.