Havel–Hakimi algorithm

The Havel–Hakimi algorithm is an algorithm in graph theory solving the graph realization problem. That is, it answers the following question: Given a finite list of nonnegative integers in non-increasing order, is there a simple graph such that its degree sequence is exactly this list? A simple graph contains no double edges or loops.^[1] The degree sequence is a list of numbers in nonincreasing order indicating the number of edges incident to each vertex in the graph.^[2] If a simple graph exists for exactly the given degree sequence, the list of integers is called graphic. The Havel-Hakimi algorithm constructs a special solution if a simple graph for the given degree sequence exists, or proves that one cannot find a positive answer. This construction is based on a recursive algorithm. The algorithm was published by Havel (1955), and later by Hakimi (1962).

The algorithm

The Havel-Hakimi algorithm is based on the following theorem.

Let ${\displaystyle A=(s,t_{1},...,t_{s},d_{1},...,d_{n})}$ be a finite list of nonnegative integers that is nonincreasing. Let ${\displaystyle A'=(t_{1}-1,...,t_{s}-1,d_{1},...,d_{n})}$ be a second finite list of nonnegative integers that is rearranged to be nonincreasing. List ${\displaystyle A}$ is graphic if and only if list ${\displaystyle A'}$ is graphic.

If the given list ${\displaystyle A}$ is graphic, then the theorem will be applied at most ${\displaystyle n-1}$ times setting in each further step ${\displaystyle A:=A'}$. Note that it can be necessary to sort this list again. This process ends when the whole list ${\displaystyle A'}$ consists of zeros. Let ${\displaystyle G}$ be a simple graph with the degree sequence ${\displaystyle A}$: Let the vertex ${\displaystyle S}$ have degree ${\displaystyle s}$; let the vertices ${\displaystyle T_{1},...,T_{s}}$ have respective degrees ${\displaystyle t_{1},...,t_{s}}$; let the vertices ${\displaystyle D_{1},...,D_{n}}$ have respective degrees ${\displaystyle d_{1},...,d_{n}}$. In each step of the algorithm, one constructs the edges of a graph with vertices ${\displaystyle T_{1},...,T_{s}}$—i.e., if it is possible to reduce the list ${\displaystyle A}$ to ${\displaystyle A'}$, then we add edges ${\displaystyle \{S,T_{1}\},\{S,T_{2}\},\cdots ,\{S,T_{s}\}}$. When the list ${\displaystyle A}$ cannot be reduced to a list ${\displaystyle A'}$ of nonnegative integers in any step of this approach, the theorem proves that the list ${\displaystyle A}$ from the beginning is not graphic.

Proof

The following is a summary based on the proof of the Havel-Hakimi algorithm in Invitation to Combinatorics (Shahriari 2022).

To prove the Havel-Hakimi algorithm always works, assume that ${\displaystyle A'}$ is graphic, and there exists a simple graph ${\displaystyle G'}$ with the degree sequence ${\displaystyle A'=(t_{1}-1,...,t_{s}-1,d_{1},...,d_{n})}$. Then we add a new vertex ${\displaystyle v}$ adjacent to the ${\displaystyle s}$ vertices with degrees ${\displaystyle t_{1}-1,...,t_{s}-1}$ to obtain the degree sequence ${\displaystyle A}$.

To prove the other direction, assume that ${\displaystyle A}$ is graphic, and there exists a simple graph ${\displaystyle G}$ with the degree sequence ${\displaystyle A=(s,t_{1},...,t_{s},d_{1},...,d_{n})}$ and vertices ${\displaystyle S,T_{1},...,T_{s},D_{1},...,D_{n}}$. We do not know which ${\displaystyle s}$ vertices are adjacent to ${\displaystyle S}$, so we have two possible cases.

In the first case, ${\displaystyle S}$ is adjacent to the vertices ${\displaystyle T_{1},...,T_{s}}$ in ${\displaystyle G}$. In this case, we remove ${\displaystyle S}$ with all its incident edges to obtain the degree sequence ${\displaystyle A'}$.

In the second case, ${\displaystyle S}$ is not adjacent to some vertex ${\displaystyle T_{i}}$ for some ${\displaystyle 1\leq i\leq s}$ in ${\displaystyle G}$. Then we can change the graph ${\displaystyle G}$ so that ${\displaystyle S}$ is adjacent to ${\displaystyle T_{i}}$ while maintaining the same degree sequence ${\displaystyle A}$. Since ${\displaystyle S}$ has degree ${\displaystyle s}$, the vertex ${\displaystyle S}$ must be adjacent to some vertex ${\displaystyle D_{j}}$ in ${\displaystyle G}$ for ${\displaystyle 1\leq j\leq n}$: Let the degree of ${\displaystyle D_{j}}$ be ${\displaystyle d_{j}}$. We know ${\displaystyle t_{i}\geq d_{j}}$, as the degree sequence ${\displaystyle A}$ is in non-increasing order.

Since ${\displaystyle t_{i}\geq d_{j}}$, we have two possibilities: Either ${\displaystyle t_{i}=d_{j}}$, or ${\displaystyle t_{i}>d_{j}}$. If ${\displaystyle t_{i}=d_{j}}$, then by switching the places of the vertices ${\displaystyle T_{i}}$ and ${\displaystyle D_{j}}$, we can adjust ${\displaystyle G}$ so that ${\displaystyle S}$ is adjacent to ${\displaystyle T_{i}}$ instead of ${\displaystyle D_{j}.}$ If ${\displaystyle t_{i}>d_{j}}$, then since ${\displaystyle T_{i}}$ is adjacent to more vertices than ${\displaystyle D_{j}}$, let another vertex ${\displaystyle W}$ be adjacent to ${\displaystyle T_{i}}$ and not ${\displaystyle D_{j}}$. Then we can adjust ${\displaystyle G}$ by removing the edges ${\displaystyle \left\{S,D_{j}\right\}}$ and ${\displaystyle \left\{T_{i},W\right\}}$, and adding the edges ${\displaystyle \left\{S,T_{i}\right\}}$ and ${\displaystyle \left\{W,D_{j}\right\}}$. This modification preserves the degree sequence of ${\displaystyle G}$, but the vertex ${\displaystyle S}$ is now adjacent to ${\displaystyle T_{i}}$ instead of ${\displaystyle D_{j}}$. In this way, any vertex not connected to ${\displaystyle S}$ can be adjusted accordingly so that ${\displaystyle S}$ is adjacent to ${\displaystyle T_{i}}$ while maintaining the original degree sequence ${\displaystyle A}$ of ${\displaystyle G}$. Thus any vertex not connected to ${\displaystyle S}$ can be connected to ${\displaystyle S}$ using the above method, and then we have the first case once more, through which we can obtain the degree sequence ${\displaystyle A'}$. Hence, ${\displaystyle A}$ is graphic if and only if ${\displaystyle A'}$ is also graphic.

Examples

Let ${\displaystyle 6,3,3,3,3,2,2,2,2,1,1}$ be a nonincreasing, finite degree sequence of nonnegative integers. To test whether this degree sequence is graphic, we apply the Havel-Hakimi algorithm:

First, we remove the vertex with the highest degree — in this case, ${\displaystyle 6}$ —  and all its incident edges to get ${\displaystyle 2,2,2,2,1,1,2,2,1,1}$ (assuming the vertex with highest degree is adjacent to the ${\displaystyle 6}$ vertices with next highest degree). We rearrange this sequence in nonincreasing order to get ${\displaystyle 2,2,2,2,2,2,1,1,1,1}$. We repeat the process, removing the vertex with the next highest degree to get ${\displaystyle 1,1,2,2,2,1,1,1,1}$ and rearranging to get ${\displaystyle 2,2,2,1,1,1,1,1,1}$. We continue this removal to get ${\displaystyle 1,1,1,1,1,1,1,1}$, and then ${\displaystyle 0,0,0,0,0,0,0,0}$. This sequence is clearly graphic, as it is the simple graph of ${\displaystyle 8}$ isolated vertices.

To show an example of a non-graphic sequence, let ${\displaystyle 6,5,5,4,3,2,1}$ be a nonincreasing, finite degree sequence of nonnegative integers. Applying the algorithm, we first remove the degree ${\displaystyle 6}$ vertex and all its incident edges to get ${\displaystyle 4,4,3,2,1,0}$. Already, we know this degree sequence is not graphic, since it claims to have ${\displaystyle 6}$ vertices with one vertex not adjacent to any of the other vertices; thus, the maximum degree of the other vertices is ${\displaystyle 4}$. This means that two of the vertices are connected to all the other vertices with the exception of the isolated one, so the minimum degree of each vertex should be ${\displaystyle 2}$; however, the sequence claims to have a vertex with degree ${\displaystyle 1}$. Thus, the sequence is not graphic.

For the sake of the algorithm, if we were to reiterate the process, we would get ${\displaystyle 3,2,1,0,0}$ which is yet more clearly not graphic. One vertex claims to have a degree of ${\displaystyle 3}$, and yet only two other vertices have neighbors. Thus the sequence cannot be graphic.

See also

• Erdős–Gallai theorem

Notes

1. From Shahriari (2022, p. 48): "Definition 2.17 (Graphs & Subgraphs). A simple graph (or just a graph) G is a pair of sets (V, E) where V is a nonempty set called the set of vertices of G, and E is a (possibly empty) set of unordered pairs of distinct elements of V. The set E is called the set of edges of G. If the number of vertices of G is finite, then G is a finite graph (or a finite simple graph)."
2. From Shahriari (2022, p. 355): "Definition 10.6 (The degree sequence of a graph; Graphic sequences). The degree sequence of a graph is the list of the degrees of its vertices in non-increasing order. A non-increasing sequence of non-negative integers is called graphic, if there exists a simple graph whose degree sequence is precisely that sequence.”

References

• Havel, Václav (1955), "A remark on the existence of finite graphs", Časopis pro pěstování matematiky (in Czech), 80 (4): 477–480, doi:10.21136/CPM.1955.108220
• Hakimi, S. L. (1962), "On realizability of a set of integers as degrees of the vertices of a linear graph. I", Journal of the Society for Industrial and Applied Mathematics, 10 (3): 496–506, doi:10.1137/0110037, MR 0148049.
• Shahriari, Shahriar (2022), Invitation to Combinatorics, Cambridge U. Press.
• West, Douglas B. (2001). Introduction to graph theory. Second Edition. Prentice Hall, 2001. 45-46.