Strong Subgraph Connectivity of Digraphs

Abstract

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