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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
##### Keywords:
subgraphs; k-trees; partial 3-tree
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##### References:
 [1] Arnborg, SIAM J. Alg. Discr. Math. 7 pp 305– (1986) [2] Arnborg, SIAM J. Alg. Discr. Math. 8 pp 277– (1987) [3] Beineke, Mathematika 18 pp 141– (1971) [4] Dirac, Abh. Math. Sem. Univ. Hamburg 25 pp 71– (1961) [5] Duffin, J. Math. Appl. 10 pp 302– (1965) [6] Graph Theory, Addison-Wesley, Reading, MA (1972). [7] , and , A characterization of the partial 3-tree in terms of certain structures. In Proceedings of ISCAS 1985, IEEE. [8] Rose, J. Math. Anal. Appl. 32 pp 597– (1970) [9] Rose, Discre. Math. 7 pp 317– (1974) · Zbl 0285.05128 · doi:10.1016/0012-365X(74)90042-9 [10] Connectivity in Graphs, Toronto University Press, Toronto (1966). [11] Wald, Networks 13 pp 159– (1983)
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