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On the representation of triangulation graphs in trees. (English) Zbl 0541.05049
A graph G is said to be tree-representable if there exist a tree T and a family \(T_ i\) (\(i\in I)\) of subtrees of T such that G is isomorphic to the intersection graph of this family. It has been shown independently by several authors that if G is a finite graph then G is tree-representable if and only if G is chordal, i.e., no cycle of length at least 4 is an induced subgraph of G. It is not true that every infinite chordal graph is tree-representable. The author’s main result is a characterization of all tree-representable graphs, given by means of simplicial decompositions.
Reviewer: L.Lesniak

05C75 Structural characterization of families of graphs
05C05 Trees
Full Text: DOI
[1] Buneman, P., A characterization of rigid circuit graphs, Discrete math., 9, 205-212, (1974) · Zbl 0288.05128
[2] Dirac, G.A., On rigid circuit graphs, Abh. math. sem. univ. Hamburg, 25, 71-76, (1961) · Zbl 0098.14703
[3] Gavril, F., The intersection graphs of subtrees in trees are exactly the chordal graphs, J. combin. theory B, 16, 47-56, (1974) · Zbl 0266.05101
[4] Golumbic, M.C., Algorithmic graph theory and perfect graphs, (1980), Academic Press New York · Zbl 0541.05054
[5] Gyárfas, A.; Lehel, J., A Helly type problem in trees, (), 571-584, Coll. Math. Soc. J. Bolyai 4, North-Holland, Amsterdam
[6] Halin, R., Über simpliziale zerfällungen beliebiger (endlicher oder unendlicher) graphen, Math. ann., 156, 216-225, (1964) · Zbl 0125.11605
[7] Halin, R., Über unendliche wege in graphen, Math. ann., 157, 125-137, (1964) · Zbl 0125.11701
[8] Halin, R., Graphentheorie II, (1981), Wissenschaftliche Buchgesellschaft Darmstadt
[9] Walter, J.R., Representations of chordal graphs as subtrees of a tree, J. graph theory, 2, 265-267, (1978) · Zbl 0441.05022
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