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- W1839390018 abstract "A spanning tree with no vertices of degree 2 is called a Homeomorphically irreducible spanning tree,(HIST). Based on a HIST embedded in the plane, a Halin graph is formed by connecting the leaves of the tree into a cycle following the cyclic order determined by the embedding. Both of the determination problems of whether a graph contains a HIST or whether a graph contains a spanning Halin graph are shown to be NP-complete. It was conjectured by Albertson, Berman, Hutchinson, and Thomassen in 1990 that a {it every surface triangulation of at least four vertices contains a HIST},(confirmed). And it was conjectured by Lov'asz and Plummer that {it every 4-connected plane triangulation contains a spanning Halin graph},(disproved). Balancing the above two facts, in this paper, we consider generalized Halin graphs, a family of graph structures which are stronger than HISTs but weaker than Halin graphs in the sense of their construction constraints. To be exact, a generalized Halin graph is formed from a HIST by connecting its leaves into a cycle. Since a generalized Halin graph needs not to be planar, we investigate the minimum degree condition for a graph to contain it as a spanning subgraph. We show that there exists a positive integer $n_0$ such that any 3-connected graph with $nge n_0$ vertices and minimum degree at least $(2n+3)/5$ contains a spanning generalized Halin graph. As an application, the result implies that under the same condition, the graph $G$ contains a wheel-minor of order at least $n/2$. The minimum degree condition in the result is best possible." @default.
- W1839390018 created "2016-06-24" @default.
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- W1839390018 date "2014-12-07" @default.
- W1839390018 modified "2023-09-24" @default.
- W1839390018 title "Minimum degree condition for spanning generalized Halin graphs" @default.
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- W1839390018 doi "https://doi.org/10.48550/arxiv.1412.2374" @default.
- W1839390018 hasPublicationYear "2014" @default.
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