2013/07/08 by Liliana Alcón, Alcón, Liliana, Marisa Gutiérrez +3
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #Complexity and Algorithms in Graphs #FOS: Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.1307.2139
openalex publication_date 2013/07/08 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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 by 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, are the minimal forbidden induced subgraphs for [h,2,1], with h≥ 3. Notice that by taking h=3 we obtain a characterization by minimal forbidden induced subgraphs of the class VPT ∩ EPT=EPT ∩ Chordal=[3,2,2]=[3,2,1].