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Graphs with two trivial critical ideals

Authors
Publisher
Elsevier B.V.
Publication Date
Volume
167
Identifiers
DOI: 10.1016/j.dam.2013.11.002
Keywords
  • Critical Ideal
  • Critical Group
  • Generalized Laplacian Matrix
  • Forbidden Induced Subgraph

Abstract

Abstract The critical ideals of a graph are the determinantal ideals of the generalized Laplacian matrix associated to a graph. A basic property of the critical ideals of graphs asserts that the graphs with at most k trivial critical ideals, Γ≤k, are closed under induced subgraphs. In this article we find the set of minimal forbidden subgraphs for Γ≤2, and we use this forbidden subgraphs to get a classification of the graphs in Γ≤2. As a consequence we give a classification of the simple graphs whose critical group has two invariant factors equal to one. At the end of this article we give two infinite families of forbidden subgraphs.

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