Analysis of the Lifting Graph

TL;DR

This paper provides a comprehensive analysis of the structure of the lifting graph, a key concept in graph connectivity and network design, building on decades of research and applications.

Contribution

It offers a detailed structural analysis of the lifting graph and its complement, advancing understanding for connectivity augmentation and graph orientation problems.

Findings

Structural results on the non-admissibility graph

Applications to connectivity augmentation

Insights into infinite graph orientations

Abstract

The `lifting` or `splitting-off` operation on graphs is performed by deleting two edges sv and sw having a common end s and adding a new edge between v and w. Such a lift is considered good if it preserves a certain local edge-connectivity between the pairs of vertices different from the vertex s at which lifting takes place. The operation is important for inductive proofs concerning edge-connectivity, and can be seen widely applied in the literature on connectivity augmentation, network design, orientation (of finite and infinite graphs), and edge-disjoint linkage. It was studied by Lovasz, who used the term splitting-off, and Mader, who used the term lifting. They proved the first two significant results on it, in 1976 and 1978 respectively, showing the existence of a good lift under certain conditions. Then it was used and studied by other researchers, through the 1980s, both for undirected and directed graphs. In particular, it was investigated further by Frank who proved in 1992 that there are floor of deg(s)/2 disjoint good lifts. Motivated by the applications, a new method for studying the operation was introduced by Jordan in the late 1990s. He defined and studied the structure of the `non-admissibility` graph, which is the complement of the lifting graph; the subject of this paper. He proved a number of significant structural results on it, which he applied to connectivity augmentation. Independently, in 2016, Thomassen defined the `lifting graph`, and called its complement the `bad graph`, to apply it in finding orientations of infinite graphs. Later in the same year, Thomassen with Ok and Richter extended the study, and applied their results to linkages in infinite graphs. Here we give a more comprehensive analysis of the structure of the lifting graph.