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- W4310922022 abstract "The lifting graph $L(G,s,k)$ is an auxiliary graph, useful in edge-connectivity proofs, defined via an operation called textit{lifting} performed on pairs of edges incident with a vertex $s$ in a graph $G$. It encodes the information of whether a pair of edges incident with $s$ forms a good lift in the sense that lifting them preserves a certain prescribed local edge-connectivity $k$ between the pairs of vertices different from $s$. Lifting was studied by Lov'asz cite{Lovasz} and Mader and they proved the first two significant results about it in 1976 and 1978 respectively, showing the existence of one good lift under certain conditions. Later in 1992 lifting was investigated further by Frank, who proved that there are $lfloor deg(s)/2 rfloor $ many good lifts. Thomassen showed in 2016 that if $G$ is Eulerian, then the complement of $L(G,s,k)$ is disconnected. This was extended in the same year by Ok, Richter, and Thomassen to graphs $G$ where only $deg(s)$ and $k$ are required to be even, and they began the first detailed study of the structure of the lifting graph for a general $G$. Here we give a more comprehensive analysis of the structure of the lifting graph $L(G,s,k)$, studied through the interaction of its maximal independent sets, which is a new approach revealing more of the structure than seen in the inductive proof by Ok, Richter, and Thomassen. We prove specific structures for the lifting graph depending on the different possibilities for the parities of $deg(s)$ and $k$. An important consequence is the description of the structure of a graph in case its lifting graph has a connected complement. The graph $G-s$ consists of a path or cycle of $((k+1)/2)$-edge-connected blobs, with consecutive blobs joined by precisely $(k-1)/2$ edges except in the case when the structure is a path of length $2$ of blobs." @default.
- W4310922022 created "2022-12-21" @default.
- W4310922022 creator A5057519666 @default.
- W4310922022 date "2022-12-06" @default.
- W4310922022 modified "2023-10-15" @default.
- W4310922022 title "Analysis of the Lifting Graph" @default.
- W4310922022 doi "https://doi.org/10.48550/arxiv.2212.03347" @default.
- W4310922022 hasPublicationYear "2022" @default.
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