On a Conjecture Concerning Helly Circle Graphs

We say that G is an e-circle graph if there is a bijection between its vertices and straight lines on the cartesian plane such that two vertices are adjacent in G if and only if the corresponding lines intersect inside the circle of radius one. This definition suggests a method for deciding whether a given graph G is an e-circle graph, by constructing a convenient system S of equations and inequations which represents the structure of G, in such a way that G is an e-circle graph if and only if S has a solution. In fact, e-circle graphs are exactly the circle graphs (intersection graphs of chords in a circle), and thus this method provides an analytic way for recognizing circle graphs. Departamento de Computación, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria (1428), Buenos Aires, Argentina. Partially supported by UBACyT Grants X036 and X127, and PID Conicet Grant 644/98, Argentina. E-mail: [email protected] Departamento de Computación, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria (1428), Buenos Aires, Argentina. Partially supported by UBACyT Grant X127, Argentina. E-mail: [email protected] Departamento de Computación, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria (1428), Buenos Aires, Argentina. Partially supported by UBACyT Grant X036, Argentina. E-mail: [email protected] NCE, Universidade Federal do Rio de Janeiro, RJ, Caixa Postal 2324, 20001-970, Brasil. Partially suported by Conselho Nacional de Desenvolvimento Cient́ıfico e Tecnológico, CNPq Brasil. E-mail: [email protected] Universidade Federal do Rio de Janeiro, Instituto de Matemática, NCE and COPPE, Caixa Postal 2324, 20001-970, Rio de Janeiro, RJ, Brasil. Partially supported by the Conselho Nacional de Desenvolvimento Cient́ıfico e Tecnológico CNPq, and Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro FAPERJ, Brasil. E-mail: [email protected]