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Our interest in the class of k-trees and their partial graphs and subgraphs is motivated by some practical questions about the reliability of communication networks in the presence of constrained line- and site-failures, and about the complexity of queries in a data base system. We have found a set of confluent graph reductions such that any graph can be reduced to the empty graph if and only if it is a subgraph of a 3-tree. This set of reductions yields a polynomial time algorithm for deciding if a given graph is a partial 3-tree and for finding one of its embeddings in a 3-tree when such an embedding exists. Our result generalizes a previously known recognition algorithm for partial 2-trees (series-parallel graphs).
Arnborg et al. (Tue,) studied this question.