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Absolute Retracts and Varieties of Reflexive Graphs

Published online by Cambridge University Press:  20 November 2018

Pavol Hell  and
Ivan Rival
Pavol Hell
Affiliation:
Simon Fraser University, Burnaby, British Columbia
Ivan Rival
Affiliation:
The University of Ottawa, Ottawa, Ontario
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For a graph G, let V(G) denote its vertex set and E(G) its edge set. Here we shall only consider reflexive graphs, that is graphs in which every vertex is adjacent to itself. These adjacencies, i.e., the loops, will not be depicted in the figures, although we always assume them present. For graphs G and H, an edge-preserving map (or homomorphism) of G to H is a mapping of V(G) to V(H) such that f(g) is adjacent to f(g′) in H whenever g is adjacent to g′ in G. Because our graphs are reflexive, an edge-preserving map can identify adjacent vertices, i.e., possibly f(g) = f(g′) for some g adjacent to g′, cf. Figure 1(a).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 3 , 01 June 1987 , pp. 544 - 567
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Berge, C., Graphs and hyper graphs (North Holland, Amsterdam, London, 1973).Google Scholar
2. Borsuk, K., Sur les rétractes, Fundamenta Math. 17 (1931), 152–170.Google Scholar
3. Duffus, D. and Rival, I., A structure theory for ordered sets, Discrete Math 35 (1981), 53–118.Google Scholar
4. Hell, P., Rétractions de graphes, Ph.D. thesis, Université de Montréal (1972).Google Scholar
5. Hell, P., Absolute retracts in graphs in Graphs and combinatorics, Springer-Verlag, Lect. Notes Math. 406 (1973), 291–301.CrossRefGoogle Scholar
6. Hell, P., Graph retractions, Atti dei convegni lincei 17 (1976), 263–268.Google Scholar
7. Jawhari, E. M., Pouzet, M. and Rival, I., A classification of reflexive graphs: the use of “holes”, Can. J. Math. 38 (1986), 1299–1328.Google Scholar
8. Nowakowski, R. J. and Rival, I., Fixed-edge theorem for graphs with loops, J. Graph Theory 3 (1979), 339–350.Google Scholar
9. Nowakowski, R. J. and Rival, I., The smallest graph variety containing all paths, Discrete Math. 43 (1983), 223–234.Google Scholar
10. Pesch, E., Absolute Retrakte von Graphen, Diplomarbeit, Technische Hochschule Darmstadt (1982).Google Scholar
11. Pesch, E. and Poguntke, W., A characterization of absolute retracts of n-chromatic graphs, Discrete Math. 57 (1985), 99–104.Google Scholar
12. Quilliot, A., Homomorphismes, points fixes, rétractions et jeux de poursuite dans les graphes, let ensembles ordonnées et les espaces métriques, Thèse d'Etat, Université de Paris VI (1983).Google Scholar
13. Quilliot, A., An application of the Helly property to the partially ordered sets, J. Combin. Theory A 35 (1983), 185–198.Google Scholar
14. Rosenfeld, M., On a problem of C. E. Shannon in graph theory, Proc. Amer. Math. Soc. 18 (1967), 315–319.Google Scholar
15. Rosenfeld, M., Sabidussi, Personal communication (1970).Google Scholar