Spanning trees of smallest maximum degree in subdivisions of graphs

10/10/2022

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by   Christoph Brause, et al.

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Given a graph G and a positive integer k, we study the question whether G^⋆ has a spanning tree of maximum degree at most k where G^⋆ is the graph that is obtained from G by subdividing every edge once. Using matroid intersection, we obtain a polynomial algorithm for this problem and a characterization of its positive instances. We use this characterization to show that G^⋆ has a spanning tree of bounded maximum degree if G is contained in some particular graph class. We study the class of 3-connected graphs which are embeddable in a fixed surface and the class of (p-1)-connected K_p-minor-free graphs for a fixed integer p. We also give tightness examples for most of these classes.