We solve Open Problem (xvi) from Perfect Problems of Chvátal [1] available at ftp://dimacs.rutgers.edu/pub/perfect/problems.tex: Is there a class C of perfect graphs such that (a) C does not include all perfect graphs and (b) every perfect graph contains a vertex whose neighbors induce a subgraph that belongs to C? A class P is called locally reducible if there exists a proper subclass C of P such that every graph in P contains a local subgraph belonging to C. We characterize locally reducible hereditary classes. It implies that there are infinitely many solutions to Open Problem (xvi). However, it is impossible to find a hereditary class C of perfect graphs satisfying both (a) and (b).
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1318,
author = {Igor E. Zverovich},
title = {On a perfect problem},
journal = {Discussiones Mathematicae Graph Theory},
volume = {26},
year = {2006},
pages = {273-277},
zbl = {1142.05033},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1318}
}
Igor E. Zverovich. On a perfect problem. Discussiones Mathematicae Graph Theory, Tome 26 (2006) pp. 273-277. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1318/
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