Steiner Type Packing Problems in Digraphs: A Survey

TL;DR

This survey catalogs the landscape of directed Steiner type packing problems in digraphs, covering directed Steiner tree/path/pendant packings, strong subgraph/arc decompositions, and directed Steiner cycle packings. It maps the algorithmic and complexity frontier across general, symmetric, Eulerian, semicomplete, and composed digraph classes, linking to foundational results such as Menger's theorem, Edmonds' branching, and cycle/ Hamiltonian decompositions. The work highlights NP-hardness and inapproximability boundaries, identifies polynomial-time solvable regimes, and discusses conjectures and open problems that guide future work in digraph packing theory. It also situates directed Steiner type packing within broader connectivity notions, including directed path/ cycle connectivity and strong subgraph connectivity, to illuminate structural principles underlying packing problems.

Abstract

Graph packing problem is one of the central problems in graph theory and combinatorial optimization. The famous Steiner tree packing problem in undirected graphs has become an well-established area. It is natural to extend this problem to digraphs, and such problems in digraphs are called directed Steiner type packing problems. In this survey we overview known results on several directed Steiner type packing problems. The paper is divided into seven sections: introduction, directed Steiner tree packing problem, directed Steiner path packing problem, directed pendant Steiner tree packing problem, strong subgraph packing problem, strong arc decomposition problem, directed Steiner cycle packing problem. This survey also contains some conjectures and open problems for further study.

Figures (1)

• Figure 1: Digraph $S_4$

Theorems & Definitions (72)

• Theorem 1.1
• Conjecture 1.1
• Conjecture 1.2
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6