Geodesic
On a characterization of $k$-trees
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 361-365

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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.
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.
DOI : 10.1007/s10587-015-0180-7
Classification : 05C05
Keywords: characterization; $k$-tree; $K_t$-minor 
Zeng, De-Yan; Yin, Jian-Hua. On a characterization of $k$-trees. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 361-365. doi: 10.1007/s10587-015-0180-7
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