Original Paper

Graphs and Combinatorics

, Volume 30, Issue 4, pp 977-1002

First online:

Graph Invertibility

  • Cam McLemanAffiliated withDepartment of Mathematics, The University of Michigan-Flint Email author 
  • , Erin McNicholasAffiliated withDepartment of Mathematics, Willamette University

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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