Factor-critical graph

A factor-critical graph, together with perfect matchings of the subgraphs formed by removing one of its vertices.

In graph theory, a mathematical discipline, a factor-critical graph (or hypomatchable graph[1][2]) is a graph with n vertices in which every induced subgraph of n − 1 vertices has a perfect matching. (A perfect matching in a graph is a subset of its edges with the property that each of its vertices is the endpoint of exactly one of the edges in the subset.)

A matching that covers all but one vertex of a graph is called a near-perfect matching. So equivalently, a factor-critical graph is a graph in which there are near-perfect matchings that avoid every possible vertex.

  1. ^ Cite error: The named reference pe74 was invoked but never defined (see the help page).
  2. ^ Cornuéjols, G.; Pulleyblank, W. R. (1983), "Critical graphs, matchings and tours or a hierarchy of relaxations for the travelling salesman problem", Combinatorica, 3 (1): 35–52, doi:10.1007/BF02579340, MR 0716420, S2CID 35825797.