Abstract
A graph G is a k-tree if either G is the complete graph on k + 1 vertices, or G has a vertex v whose neighborhood is a clique of order k and the graph obtained by removing v from G is also a k-tree. Clearly, a k-tree has at least k + 1 vertices, and G is a 1-tree (usual tree) if and only if it is a 1-connected graph and has no K 3-minor. In this paper, motivated by some properties of 2-trees, we obtain a characterization of k-trees as follows: if G is a graph with at least k + 1 vertices, then G is a k-tree if and only if G has no K k+2-minor, G does not contain any chordless cycle of length at least 4 and G is k-connected.
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This work is supported by National Natural Science Foundation of China (Grant No. 11161016).
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Zeng, DY., Yin, JH. On a characterization of k-trees. Czech Math J 65, 361–365 (2015). https://doi.org/10.1007/s10587-015-0180-7
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DOI: https://doi.org/10.1007/s10587-015-0180-7