On a perfect problem
Discussiones Mathematicae Graph Theory (2006)
- Volume: 26, Issue: 2, page 273-277
- ISSN: 2083-5892
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topIgor E. Zverovich. "On a perfect problem." Discussiones Mathematicae Graph Theory 26.2 (2006): 273-277. <http://eudml.org/doc/270312>.
@article{IgorE2006,
abstract = {
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).
},
author = {Igor E. Zverovich},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {hereditary classes; perfect graphs},
language = {eng},
number = {2},
pages = {273-277},
title = {On a perfect problem},
url = {http://eudml.org/doc/270312},
volume = {26},
year = {2006},
}
TY - JOUR
AU - Igor E. Zverovich
TI - On a perfect problem
JO - Discussiones Mathematicae Graph Theory
PY - 2006
VL - 26
IS - 2
SP - 273
EP - 277
AB -
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).
LA - eng
KW - hereditary classes; perfect graphs
UR - http://eudml.org/doc/270312
ER -
References
top- [1] V. Chvátal, Perfect Problems, available at http://www.cs.concordia.ca/∼chvatal/perfect/problems.html.
- [2] G.A. Dirac, On rigid circuit graphs, Abh. Math. Semin. Univ. Hamburg 25 (1961) 71-76, doi: 10.1007/BF02992776. Zbl0098.14703
- [3] Perfect graphs, Edited by J.L. Ramírez Alfonsín and B.A. Reed, Wiley-Interscience Series in Discrete Mathematics and Optimization (John Wiley & Sons, Ltd., Chichester, 2001) xxii+362 pp.
- [4] A.A. Zykov, Fundamentals of Graph Theory (Nauka, Moscow, 1987) 382 pp. (in Russian), translated and edited by L. Boron, C. Christenson and B. Smith (BCS Associates, Moscow, ID, 1990) vi+365 pp.
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