Characterizing intersection graphs of substars of a star by forbidden subgraphs

Starlike graphs are the intersection graphs of substars of a star. We describe characterizations by forbidden subgraphs for starlike graphs and for a special subclass of it

Asteroidal quadruples in non rooted path graphs

A directed path graph is the intersection graph of a family of directed subpaths of a directed tree. A rooted path graph is the intersection graph of a family of directed subpaths of a rooted tree. Rooted path graphs are directed path graphs. Several characterizations are known for directed path graphs: one by forbidden induced subgraphs and one by forbidden asteroids. It is an open problem to find such characterizations for rooted path graphs. For this purpose, we are studying in this paper directed path graphs that are non rooted path graphs. We prove that such graphs always contain an asteroidal quadruple.Facultad de Ciencias Exacta

Diferencia entre las clases UV y DV

Los grafos cordales, utilizados para modelar problemas en Biología, fueron definidos como aquellos que no poseen ciclos inducidos de 4 o mas vértices. En este trabajo se prueba que los grafos UV no DV minimales son los soles impares; utilizando árboles cliques y la caracterización de Monma y Wei. De aquí, se tiene entonces otra forma de obtener la familia de grafos no DV minimales a partir de la familia de los no UV minimales.Facultad de Ciencias Exacta

Diferencia entre las clases UV y DV

Los grafos cordales, utilizados para modelar problemas en Biología, fueron definidos como aquellos que no poseen ciclos inducidos de 4 o mas vértices. En este trabajo se prueba que los grafos UV no DV minimales son los soles impares; utilizando árboles cliques y la caracterización de Monma y Wei. De aquí, se tiene entonces otra forma de obtener la familia de grafos no DV minimales a partir de la familia de los no UV minimales.Facultad de Ciencias Exacta

Diferencia entre las clases UV y DV

Los grafos cordales, utilizados para modelar problemas en Biología, fueron definidos como aquellos que no poseen ciclos inducidos de 4 o mas vértices. En este trabajo se prueba que los grafos UV no DV minimales son los soles impares; utilizando árboles cliques y la caracterización de Monma y Wei. De aquí, se tiene entonces otra forma de obtener la familia de grafos no DV minimales a partir de la familia de los no UV minimales.Facultad de Ciencias Exacta

Asteroidal quadruples in non rooted path graphs

A directed path graph is the intersection graph of a family of directed subpaths of a directed tree. A rooted path graph is the intersection graph of a family of directed subpaths of a rooted tree. Rooted path graphs are directed path graphs. Several characterizations are known for directed path graphs: one by forbidden induced subgraphs and one by forbidden asteroids. It is an open problem to find such characterizations for rooted path graphs. For this purpose, we are studying in this paper directed path graphs that are non rooted path graphs. We prove that such graphs always contain an asteroidal quadruple.Facultad de Ciencias Exacta

On rooted directed path graphs

An asteroidal triple is a stable set of three vertices such that each pair is connected by a path avoiding the neighborhood of the third vertex. An asteroidal quadruple is a stable set of four vertices such that any three of them is an asteroidal triple. Two non adjacent vertices are linked by a special connection if either they have a common neighbor or they are the endpoints of two vertex-disjoint chordless paths satisfying certain technical conditions. Cameron, Ho`ang, and L´evˆeque [DIMAP Workshop on Algorithmic Graph Theory, 67–74, Electron. Notes Discrete Math., 32, Elsevier, 2009] proved that if a pair of non adjacent vertices are linked by a special connection then in any directed path model T the subpaths of T corresponding to the vertices forming the special connection have to overlap and they force T to be completely directed in one direction between these vertices. Special connections along with the concept of asteroidal quadruple play an important role to study rooted directed path graphs, which are the intersection graphs of directed paths in a rooted directed tree. In this work we define other special connections; these special connections along with the ones defined by Cameron, Ho`ang, and L´evˆeque are nine in total, and we prove that every one forces T to be completely directed in one direction between these vertices. Also, we give a characterization of rooted directed path graphs whose rooted models cannot be rooted on a bold maximal clique. As a by-product of our result, we build new forbidden induced subgraphs for rooted directed path graphs.Fil: Tondato, Silvia Beatriz. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; ArgentinaFil: Gutierrez, Marisa. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentin

Two New Characterizations of Path Graphs

Path graphs are intersection graphs of paths in a tree. We start from the characterization of path graphs by Monma and Wei [C.L.~Monma,~and~V.K.~Wei, Intersection Graphs of Paths in a Tree, J. Combin. Theory Ser. B, 41:2 (1986) 141--181] and we reduce it to some 2-colorings subproblems, obtaining the first characterization that directly leads to a polynomial recognition algorithm. Then we introduce the collection of the attachedness graphs of a graph and we exhibit a list of minimal forbidden 2-edge colored subgraphs in each of the attachedness graph.Comment: 18 pages, 6 figure