A characterization of partial 3-trees is given based on the elimination sequence of vertices. It is proved that a partial 3-tree contains a vertex set satisfying either of certain three kinds of neighborhood relations on vertices and that a graph is a partial 3-tree if and only if eliminating such a vertex set results in a partial 3-tree. These results yield anO(n 2) time algorithm to recognize 3-trees.
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Kajitani, Y., Ishizuka, A. & Ueno, S. Characterization of partial 3-trees in terms of three structures. Graphs and Combinatorics 2, 233–246 (1986). https://doi.org/10.1007/BF01788098
- Time Algorithm
- Neighborhood Relation
- Elimination Sequence