# On hereditary properties of composition graphs

• Volume: 18, Issue: 2, page 183-195
• ISSN: 2083-5892

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## Abstract

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The composition graph of a family of n+1 disjoint graphs ${H}_{i}:0\le i\le n$ is the graph H obtained by substituting the n vertices of H₀ respectively by the graphs H₁,H₂,...,Hₙ. If H has some hereditary property P, then necessarily all its factors enjoy the same property. For some sort of graphs it is sufficient that all factors ${H}_{i}:0\le i\le n$ have a certain common P to endow H with this P. For instance, it is known that the composition graph of a family of perfect graphs is also a perfect graph (B. Bollobas, 1978), and the composition graph of a family of comparability graphs is a comparability graph as well (M.C. Golumbic, 1980). In this paper we show that the composition graph of a family of co-graphs (i.e., P₄-free graphs), is also a co-graph, whereas for θ₁-perfect graphs (i.e., P₄-free and C₄-free graphs) and for threshold graphs (i.e., P₄-free, C₄-free and 2K₂-free graphs), the corresponding factors ${H}_{i}:0\le i\le n$ have to be equipped with some special structure.

## How to cite

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Vadim E. Levit, and Eugen Mandrescu. "On hereditary properties of composition graphs." Discussiones Mathematicae Graph Theory 18.2 (1998): 183-195. <http://eudml.org/doc/270387>.

abstract = {The composition graph of a family of n+1 disjoint graphs $\{H_i:0 ≤ i ≤ n\}$ is the graph H obtained by substituting the n vertices of H₀ respectively by the graphs H₁,H₂,...,Hₙ. If H has some hereditary property P, then necessarily all its factors enjoy the same property. For some sort of graphs it is sufficient that all factors $\{H_i: 0 ≤ i ≤ n\}$ have a certain common P to endow H with this P. For instance, it is known that the composition graph of a family of perfect graphs is also a perfect graph (B. Bollobas, 1978), and the composition graph of a family of comparability graphs is a comparability graph as well (M.C. Golumbic, 1980). In this paper we show that the composition graph of a family of co-graphs (i.e., P₄-free graphs), is also a co-graph, whereas for θ₁-perfect graphs (i.e., P₄-free and C₄-free graphs) and for threshold graphs (i.e., P₄-free, C₄-free and 2K₂-free graphs), the corresponding factors $\{H_i:0 ≤ i ≤ n\}$ have to be equipped with some special structure.},
author = {Vadim E. Levit, Eugen Mandrescu},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {composition graph; co-graphs; θ₁-perfect graphs; threshold graphs; composition; perfectness; comparability; permutation graph; factor graphs; composite graph},
language = {eng},
number = {2},
pages = {183-195},
title = {On hereditary properties of composition graphs},
url = {http://eudml.org/doc/270387},
volume = {18},
year = {1998},
}

TY - JOUR
AU - Vadim E. Levit
AU - Eugen Mandrescu
TI - On hereditary properties of composition graphs
JO - Discussiones Mathematicae Graph Theory
PY - 1998
VL - 18
IS - 2
SP - 183
EP - 195
AB - The composition graph of a family of n+1 disjoint graphs ${H_i:0 ≤ i ≤ n}$ is the graph H obtained by substituting the n vertices of H₀ respectively by the graphs H₁,H₂,...,Hₙ. If H has some hereditary property P, then necessarily all its factors enjoy the same property. For some sort of graphs it is sufficient that all factors ${H_i: 0 ≤ i ≤ n}$ have a certain common P to endow H with this P. For instance, it is known that the composition graph of a family of perfect graphs is also a perfect graph (B. Bollobas, 1978), and the composition graph of a family of comparability graphs is a comparability graph as well (M.C. Golumbic, 1980). In this paper we show that the composition graph of a family of co-graphs (i.e., P₄-free graphs), is also a co-graph, whereas for θ₁-perfect graphs (i.e., P₄-free and C₄-free graphs) and for threshold graphs (i.e., P₄-free, C₄-free and 2K₂-free graphs), the corresponding factors ${H_i:0 ≤ i ≤ n}$ have to be equipped with some special structure.
LA - eng
KW - composition graph; co-graphs; θ₁-perfect graphs; threshold graphs; composition; perfectness; comparability; permutation graph; factor graphs; composite graph
UR - http://eudml.org/doc/270387
ER -

## References

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1. [1] B. Bollobás, Extremal graph theory (Academic Press, London, 1978). Zbl0419.05031
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8. [8] M.C. Golumbic, Trivially perfect graphs, Discrete Math. 24 (1978) 105-107, doi: 10.1016/0012-365X(78)90178-4.
9. [9] M.C. Golumbic, Algorithmic graph theory and perfect graphs (Academic Press, London, 1980).
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11. [11] N.V.R. Mahadev and U.N. Peled, Threshold graphs and related topics (North-Holland, Amsterdam, 1995). Zbl0852.05001
12. [12] E. Mandrescu, Triangulated graph products, Anal. Univ. Galatzi (1991) 37-44.
13. [13] K.R. Parthasarathy, S.A. Choudum and G. Ravindra, Line-clique cover number of a graph, Proc. Indian Nat. Sci. Acad., Part A 41 (3) (1975) 281-293. Zbl0335.05127
14. [14] U.N. Peled, Matroidal graphs, Discrete Math. 20 (1977) 263-286.
15. [15] A. Pnueli, A. Lempel and S. Even, Transitive orientation of graphs and identification of permutation graphs, Canad. J. Math. 23 (1971) 160-175, doi: 10.4153/CJM-1971-016-5. Zbl0204.24604
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17. [17] G. Sabidussi, The composition of graphs, Duke Math. J. 26 (1959) 693-698, doi: 10.1215/S0012-7094-59-02667-5. Zbl0095.37802

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