Strong Subgraph $k$-connectivity

TL;DR

This paper extends the concept of generalized connectivity to directed graphs through strong subgraph $k$-connectivity, analyzing its computational complexity, algorithms, and bounds.

Contribution

It introduces strong subgraph $k$-connectivity for directed graphs, establishes NP-completeness, and provides polynomial algorithms for specific graph classes.

Findings

NP-complete for general directed graphs

Polynomial algorithms for semicomplete and symmetric digraphs

Provides bounds and open questions on strong subgraph $k$-connectivity

Abstract

Generalized connectivity introduced by Hager (1985) has been studied extensively in undirected graphs and become an established area in undirected graph theory. For connectivity problems, directed graphs can be considered as generalizations of undirected graphs. In this paper, we introduce a natural extension of generalized $k$-connectivity of undirected graphs to directed graphs (we call it strong subgraph $k$-connectivity) by replacing connectivity with strong connectivity. We prove NP-completeness results and the existence of polynomial algorithms. We show that strong subgraph $k$--connectivity is, in a sense, harder to compute than generalized $k$-connectivity. However, strong subgraph $k$-connectivity can be computed in polynomial time for semicomplete digraphs and symmetric digraphs. We also provide sharp bounds on strong subgraph $k$-connectivity and pose some open questions.