On the Tree-Degree of Graphs

Every graph is the edge intersection graph of subtrees of a tree. The tree-degree of a graph is the minimum maximal degree of the underlying tree for which there exists a subtree intersection model. Computing the tree-degree is NP-complete even for planar graphs, but polynomial time algorithms exist for outer-planar graphs, diamond-free graphs and chordal graphs. The number of minimal separators of graphs with bounded tree-degree is polynomial. This implies that the treewidth of graphs with bounded tree-degree can be computed efficiently, even without the model given in advance.

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