Abstract
A tree is called a caterpillar if all its leaves are adjacent to the same its path, and the path is called a spine of the caterpillar. Broersma and Tuinstra proved that if a connected graph G satisfies σ 2(G) ≥ |G| − k + 1 for an integer k ≥ 2, then G has a spanning tree having at most k leaves. In this paper we improve this result as follows. If a connected graph G satisfies σ 2(G) ≥ |G| − k + 1 and |G| ≥ 3k − 10 for an integer k ≥ 2, then G has a spanning caterpillar having at most k leaves. Moreover, if |G| ≥ 3k − 7, then for any longest path, G has a spanning caterpillar having at most k leaves such that its spine is the longest path. These three lower bounds on σ 2(G) and |G| are sharp.
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Kano, M., Yamashita, T., Yan, Z. (2013). Spanning Caterpillars Having at Most k Leaves. In: Akiyama, J., Kano, M., Sakai, T. (eds) Computational Geometry and Graphs. TJJCCGG 2012. Lecture Notes in Computer Science, vol 8296. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45281-9_9
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DOI: https://doi.org/10.1007/978-3-642-45281-9_9
Publisher Name: Springer, Berlin, Heidelberg
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