Degeneracy (graph theory)

In

Every finite forest has either an isolated vertex (incident to no edges) or a leaf vertex (incident to exactly one edge); therefore, trees and forests are 1-degenerate graphs. Every 1-degenerate graph is a forest.

Every finite planar graph has a vertex of degree five or less; therefore, every planar graph is 5-degenerate, and the degeneracy of any planar graph is at most five. Similarly, every outerplanar graph has degeneracy at most two,^[7] and the Apollonian networks have degeneracy three.

The

The degeneracy of a graph G was defined by Lick & White (1970) as the least k such that every induced subgraph of G contains a vertex with k or fewer neighbors. The definition would be the same if arbitrary subgraphs are allowed in place of induced subgraphs, as a non-induced subgraph can only have vertex degrees that are smaller than or equal to the vertex degrees in the subgraph induced by the same vertex set.

The two concepts of coloring number and degeneracy are equivalent: in any finite graph the degeneracy is just one less than the coloring number.^[10] For, if a graph has an ordering with coloring number κ then in each subgraph H the vertex that belongs to H and is last in the ordering has at most κ − 1 neighbors in H. In the other direction, if G is k-degenerate, then an ordering with coloring number k + 1 can be obtained by repeatedly finding a vertex v with at most k neighbors, removing v from the graph, ordering the remaining vertices, and adding v to the end of the order.

A third, equivalent formulation is that G is k-degenerate (or has coloring number at most k + 1) if and only if the edges of G can be oriented to form a

k-Cores

A k-core of a graph G is a

A vertex ${\displaystyle u}$ has coreness ${\displaystyle c}$ if it belongs to a ${\displaystyle c}$-core but not to any ${\displaystyle (c+1)}$-core.

The concept of a k-core was introduced to study the clustering structure of social networks^[12] and to describe the evolution of random graphs.^[13] It has also been applied in bioinformatics,^[14] network visualization,^[15] and resilience of networks in ecology.^[16] A survey of the topic, covering the main concepts, important algorithmic techniques as well as some application domains, may be found in Malliaros et al. (2019).

Bootstrap percolation is a random process studied as an epidemic model^[17] and as a model for fault tolerance for distributed computing.^[18] It consists of selecting a random subset of active cells from a lattice or other space, and then considering the k-core of the induced subgraph of this subset.^[19]

Algorithms

Matula & Beck (1983) outline an algorithm to derive the degeneracy ordering of a graph ${\displaystyle G=(V,E)}$ with vertex set V and edge set E in ${\displaystyle {\mathcal {O}}(\vert V\vert +\vert E\vert )}$ time and ${\displaystyle {\mathcal {O}}(\vert V\vert )}$ words of space, by storing vertices in a degree-indexed bucket queue and repeatedly removing the vertex with the smallest degree. The degeneracy k is given by the highest degree of any vertex at the time of its removal.

In more detail, the algorithm proceeds as follows:

• Initialize an output list L.
• Compute a number d_v for each vertex v in G, the number of neighbors of v that are not already in L. Initially, these numbers are just the degrees of the vertices.
• Initialize an array D such that D[i] contains a list of the vertices v that are not already in L for which d_v = i.
• Initialize k to 0.
• Repeat n times:
  • Scan the array cells D[0], D[1], ... until finding an i for which D[i] is nonempty.
  • Set k to max(k,i)
  • Select a vertex v from D[i]. Add v to the beginning of L and remove it from D[i].
  • For each neighbor w of v not already in L, subtract one from d_w and move w to the cell of D corresponding to the new value of d_w.

At the end of the algorithm, any vertex ${\displaystyle L[i]}$ will have at most k edges to the vertices ${\displaystyle L[1,\ldots ,i-1]}$. The l-cores of G are the subgraphs ${\displaystyle H_{l}\subset G}$ that are induced by the vertices ${\displaystyle L[1,\ldots ,i]}$, where i is the first vertex with degree ${\displaystyle \geq l}$ at the time it is added to L.

Relation to other graph parameters

If a graph G is oriented acyclically with outdegree k, then its edges may be partitioned into k

A k-vertex-connected graph is a graph that cannot be partitioned into more than one component by the removal of fewer than k vertices, or equivalently a graph in which each pair of vertices can be connected by k vertex-disjoint paths. Since these paths must leave the two vertices of the pair via disjoint edges, a k-vertex-connected graph must have degeneracy at least k. Concepts related to k-cores but based on vertex connectivity have been studied in social network theory under the name of structural cohesion.^[23]

If a graph has

The Burr–Erdős conjecture relates the degeneracy of a graph G to the Ramsey number of G, the least n such that any two-edge-coloring of an n-vertex complete graph must contain a monochromatic copy of G. Specifically, the conjecture is that for any fixed value of k, the Ramsey number of k-degenerate graphs grows linearly in the number of vertices of the graphs.^[25] The conjecture was proven by Lee (2017).

Any ${\displaystyle n}$-vertex graph with degeneracy ${\displaystyle d}$ has at most ${\textstyle (n-d)3^{d/3}}$ maximal cliques whenever ${\displaystyle d\equiv 0\mod 3}$ and ${\displaystyle n\geq d+3}$,^[26] so the class of graphs with bounded degeneracy is said to have few cliques.

Infinite graphs

Although concepts of degeneracy and coloring number are frequently considered in the context of finite graphs, the original motivation for

The degeneracy of random subsets of infinite lattices has been studied under the name of bootstrap percolation.

See also

• Graph theory
• Network science
• Percolation Theory
• Core–periphery structure
• Cereceda's conjecture

Notes

References