ISSN:
0364-9024
Content:
Results on the existence of various types of spanning subgraphs of graphs are milestones in structural graph theory and have been diversified in several directions. In the present paper, we consider “local” versions of such statements. In 1966, for instance, D. W. Barnette proved that a 3-connected planar graph contains a spanning tree of maximum degree at most 3. A local translation of this statement is that if G is a planar graph, X is a subset of specified vertices of G such that X cannot be separated in G by removing two or fewer vertices of G, then G has a tree of maximum degree at most 3 containing all vertices of X. Our results constitute a general machinery for strengthening statements about k-connected graphs (for 1 ≤ k ≤ 4) to locally spanning versions, that is, subgraphs containing a set X ⊆ V (G) of a (not necessarily planar) graph G in which only X has high connectedness. Given a graph G and X ⊆ V (G), we say M is a minor of G rooted at X, if M is a minor of G such that each bag of M contains at most one vertex of X and X is a subset of the union of all bags. We show that G has a highly connected minor rooted at X if X ⊆ V (G) cannot be separated in G by removing a few vertices of G. Combining these investigations and the theory of Tutte paths in the planar case yields locally spanning versions of six well-known results about degree-bounded trees, Hamiltonian paths and cycles, and 2-connected subgraphs of graphs.
In:
Journal of graph theory, New York, NY [u.a.] : Wiley, 1977, 105(2024), 2, Seite 209-229, 0364-9024
In:
volume:105
In:
year:2024
In:
number:2
In:
pages:209-229
Language:
English
Author information:
Mohr, Samuel 1992-
Author information:
Böhme, Thomas 1957-
Author information:
Harant, Jochen 1954-
Author information:
Schmidt, Jens M.
Author information:
Kriesell, Matthias
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