Supermodularity in Unweighted Graph Opitimization III: Highly-connected Digraphs

TL;DR

This paper characterizes degree-sequences of highly-connected simple directed graphs and solves related augmentation problems, advancing understanding of graph connectivity and degree constraints.

Contribution

It generalizes previous results to characterize degree-sequences of strongly connected digraphs and addresses degree-specified augmentation problems for directed graphs.

Findings

Characterization of degree-sequences for simple k-node-connected digraphs

Solution to directed node-connectivity augmentation with degree constraints

Solution to edge-connectivity augmentation increasing by one

Abstract

By generalizing a recent result of Hong, Liu, and Lai on characterizing the degree-sequences of simple strongly connected directed graphs, a characterization is provided for degree-sequences of simple $k$-node-connected digraphs. More generally, we solve the directed node-connectivity augmentation problem when the augmented digraph is degree-specified and simple. As for edge-connectivity augmentation, we solve the special case when the edge-connectivity is to be increased by one and the augmenting digraph must be simple.