Characterizing paths graphs on bounded degree trees by minimal forbidden induced subgraphs

An undirected graph G is called a VPT graph if it is the vertex intersection graph of a family of paths in a tree. The class of graphs which admit a VPT representation in a host tree with maximum degree at most h is denoted by [h,2,1]. The classes [h,2,1] are closed under taking induced subgraphs, therefore each one can be characterized by a family of minimal forbidden induced subgraphs. In this paper we associate the minimal forbidden induced subgraphs for [h,2,1] which are VPT with (color) h-critical graphs. We describe how to obtain minimal forbidden induced subgraphs from critical graphs, even more, we show that the family of graphs obtained using our procedure is exactly the family of VPT minimal forbidden induced subgraphs for [h,2,1]. The members of this family together with the minimal forbidden induced subgraphs for VPT (Lévêque et al., 2009; Tondato, 2009), are the minimal forbidden induced subgraphs for [h,2,1], with h ≥ 3. By taking h=3 we obtain a characterization by minimal forbidden induced subgraphs of the class V PT∩EPT=EPT∩Chordal=[3,2,2]=[3,2,1] (see Golumbic and Jamison, 1985).Facultad de Ciencias Exacta

Characterizing paths graphs on bounded degree trees by minimal forbidden induced subgraphs

An undirected graph G is called a VPT graph if it is the vertex intersection graph of a family of paths in a tree. The class of graphs which admit a VPT representation in a host tree with maximum degree at most h is denoted by [h,2,1]. The classes [h,2,1] are closed under taking induced subgraphs, therefore each one can be characterized by a family of minimal forbidden induced subgraphs. In this paper we associate the minimal forbidden induced subgraphs for [h,2,1] which are VPT with (color) h-critical graphs. We describe how to obtain minimal forbidden induced subgraphs from critical graphs, even more, we show that the family of graphs obtained using our procedure is exactly the family of VPT minimal forbidden induced subgraphs for [h,2,1]. The members of this family together with the minimal forbidden induced subgraphs for VPT (Lévêque et al., 2009; Tondato, 2009), are the minimal forbidden induced subgraphs for [h,2,1], with h ≥ 3. By taking h=3 we obtain a characterization by minimal forbidden induced subgraphs of the class V PT∩EPT=EPT∩Chordal=[3,2,2]=[3,2,1] (see Golumbic and Jamison, 1985).Facultad de Ciencias Exacta

Characterizing paths graphs on bounded degree trees by minimal forbidden induced subgraphs

An undirected graph G is called a VPT graph if it is the vertex intersection graph of a family of paths in a tree. The class of graphs which admit a VPT representation in a host tree with maximum degree at most h is denoted by [h,2,1]. The classes [h,2,1] are closed under taking induced subgraphs, therefore each one can be characterized by a family of minimal forbidden induced subgraphs. In this paper we associate the minimal forbidden induced subgraphs for [h,2,1] which are VPT with (color) h-critical graphs. We describe how to obtain minimal forbidden induced subgraphs from critical graphs, even more, we show that the family of graphs obtained using our procedure is exactly the family of VPT minimal forbidden induced subgraphs for [h,2,1]. The members of this family together with the minimal forbidden induced subgraphs for VPT (Lévêque et al., 2009; Tondato, 2009), are the minimal forbidden induced subgraphs for [h,2,1], with h ≥ 3. By taking h=3 we obtain a characterization by minimal forbidden induced subgraphs of the class V PT∩EPT=EPT∩Chordal=[3,2,2]=[3,2,1] (see Golumbic and Jamison, 1985).Fil: Alcón, Liliana Graciela. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; ArgentinaFil: Gutierrez, Marisa. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; ArgentinaFil: Mazzoleni, María Pía. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentin

Grafos ORTH[h, s, t] /

Intersection graphs are a topic that has attracted great interest from researchers in the area of graph theory since the 1960s. In this thesis we study the intersection graphs of subtrees of a tree. More precisely we will present results on the class of graphs ORTH[h, s, t]. The graphs that belong to this class are those that admit a representation by intersection of subtrees of a host tree, in which the maximum degree of the host tree is h, the maximum degree of any subtree is s, all leaves of the subtree are also leaves of the host tree and two vertices are adjacent in the graph if and only if their corresponding subtrees have at least t nodes in common and at least one of these nodes is a leaf. Results of representability, non-representability and complexity for classes ORTH[h, s, t] with the different values of the parameters h, s or t are present in this work.Grafos de interseção é um assunto que desperta grande interesse de pesquisadores da área de teoria dos grafos desde a década de 60. Nesta tese estudamos os grafos de interseção de subárvores de uma árvore. Mais precisamente vamos apresentar resultados sobre a classe de grafos ORTH[h, s, t]. Os grafos que pertencem a esta classe são aqueles que admitem uma representação por interseção de subárvores de uma árvore hospedeira, na qual o grau máximo da árvore hospedeira é h, o grau máximo de qualquer subárvore é s, todas as folhas das subárvores também são folhas da árvore hospedeira e dois vértices são adjacentes no grafo se e somente se suas subárvores correspondentes possuem no mínimo t nós em comum e ao menos um destes nós é uma folha. Resultados de representabilidade, não-representabilidade e complexidade para classes ORTH[h, s, t] com a variação do valores dos parâmetros h, s ou t estão presentes neste trabalho