Graphs and Combinatorics

, Volume 30, Issue 4, pp 977–1002 | Cite as

Graph Invertibility

  • Cam McLemanEmail author
  • Erin McNicholas
Original Paper


Extending the work of Godsil and others, we investigate the notion of the inverse of a graph (specifically, of bipartite graphs with a unique perfect matching). We provide a concise necessary and sufficient condition for the invertibility of such graphs and generalize the notion of invertibility to multigraphs. We examine the question of whether there exists a “litmus subgraph” whose bipartiteness determines invertibility. As an application of our invertibility criteria, we quickly describe all invertible unicyclic graphs. Finally, we describe a general combinatorial procedure for iteratively constructing invertible graphs, giving rise to large new families of such graphs.


Inverse graph Unique perfect matching Digraph Transitive closure Unicyclic graph 


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Copyright information

© Springer Japan 2013

Authors and Affiliations

  1. 1.Department of MathematicsThe University of Michigan-FlintFlintUSA
  2. 2.Department of MathematicsWillamette UniversitySalemUSA

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