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A characterization of partial 3-trees. (English) Zbl 0701.90092
The paper is concerned with a class of subgraphs called k-trees and their subgraphs. A k-tree is defined recursively as follows. The complete graph \(K_ k\) on k points is a k-tree. Given a k-tree G on \(n\geq k\) points, a k-tree on \(n+1\) points is obtained by adding a new point u and edges connecting u to every point of a \(K_ k\) in G. A partial k-tree is a subgraph of some k-tree. The authors establish properties of partial 3- trees and show that a graph is a partial 3-tree if and only if it has no subgraph contractible to \(K_ 5\), \(K_{2,2,2}\), \(C_ 8(1,4)\&K_ 2\times C_ 5\). Hitherto, such a characterization of partial k-trees was known only for the values of \(k\leq 2\).
Reviewer: M.Savelsbergh

MSC:
90C35 Programming involving graphs or networks
05C05 Trees
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