Either digraphs or pre-topological spaces?

ISSN: 1984 – 8625 – n´mero 6 – IFSP - Sert˜ozinho u a EITHER DIGRAPHS OR PRE-TOPOLOGICAL SPACES?
J. CARLOS S. KIIHL1 , IRWEN VALLE GUADALUPE2

RESUMO: Neste artigo mostramos que os digrafos podem ser identificados, de um modo natural, com espa¸os pre-topol´gicos finitos. Mostramos c o tamb´m que com esta identifica¸˜o a Teoria de Homotopia Regular ´ a mais e ca e apropriada a ser usada quando se trabalha com digrafos. Em particular obtemos caracteriza¸˜es gr´ficas e estruturais para algumas classes de torneios, co a mostrando a importˆncia desta nova abordagem. a Palavras-chave: Digrafos, Espa¸os Pr´-topol´gicos, Homotopia Regular c e o para Digrafos, Torneios. ABSTRACT: In this paper we show that digraphs can be identified, in a natural way, to finite pre-topological spaces. We also show that with this identification the Regular Homotopy Theory is the most appropriate one to be used when dealing with digraphs. We give some combinatorial applications of the homotopy theory of pre-topological spaces to digraphs. In particular we get structural and graphical characterizations for some classes of tournaments, showing the importance of this new approach. Keywords: Digraphs, Pre-topological spaces, Regular Homotopy for Digraphs, Tournaments.

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´ JOSE CARLOS S. KIIHL ´ Doutor em Matem´tica pela Universidade de e a Chicago, EUA. Atualmente ´ docente no IFSP, no Campus de Sert˜ozinho. Ene a dere¸o eletrˆnico: [e-mail]. c o 2 IRWEN VALLE GUADALUPE ´ Doutor em Matem´tica pela UNICAMP. Ate a ualmente est´ como docente aposentado da UNICAMP. a
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ISSN: 1984 – 8625 – n´mero 6 – IFSP - Sert˜ozinho u a 1. Introduction In Graph Theory sometimes one has to introduce certain structures in order to study or to obtain successful applications. In [24], for instance, tournaments are studied using an algebraic approach so that they are considered as algebras of a special kind. Sometimes a topological approach is used. Since graphs can always be realized in the