Degeneracy (graph theory)

In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has a vertex of degree at most k: that is, some vertex in the subgraph touches k or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of k for which it is k-degenerate. The degeneracy of a graph is a measure of how sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph.

Degeneracy is also known as the k-core number,^[1] width,^[2] and linkage,^[3] and is essentially the same as the coloring number^[4] or Szekeres–Wilf number (named after Szekeres and Wilf (1968)). k-degenerate graphs have also been called k-inductive graphs.^[5] The degeneracy of a graph may be computed in linear time by an algorithm that repeatedly removes minimum-degree vertices.^[6] The connected components that are left after all vertices of degree less than k have been (repeatedly) removed are called the k-cores of the graph and the degeneracy of a graph is the largest value k such that it has a k-core.