Kleitman–Wang algorithms
# Kleitman–Wang algorithms

The Kleitman–Wang algorithms are two different algorithms in graph theory solving the digraph realization problem, i.e. the question if there exists for a finite list of nonnegative integer pairs a simple directed graph such that its degree sequence is exactly this list. For a positive answer the list of integer pairs is called digraphic. Both algorithms construct a special solution if one exists or prove that one cannot find a positive answer. These constructions are based on recursive algorithms. Kleitman and Wang gave these algorithms in 1973.

## Kleitman–Wang algorithm (arbitrary choice of pairs)

The algorithm is based on the following theorem.

Let $S=((a_{1},b_{1}),\dots ,(a_{n},b_{n}))$ be a finite list of nonnegative integers that is in nonincreasing lexicographical order and let $(a_{i},b_{i})$ be a pair of nonnegative integers with $b_{i}>0$. List $S$ is digraphic if and only if the finite list $S'=((a_{1}-1,b_{1}),\dots ,(a_{b_{i}-1}-1,b_{b_{i}-1}),(a_{b_{i}},0),(a_{b_{i}+1},b_{b_{i}+1}),(a_{b_{i}+2},b_{b_{i}+2}),\dots ,(a_{n},b_{n}))$ has nonnegative integer pairs and is digraphic.

Note that the pair $(a_{i},b_{i})$ is arbitrarily with the exception of pairs $(a_{j},0)$. If the given list $S$ digraphic then the theorem will be applied at most $n$ times setting in each further step $S:=S'$. This process ends when the whole list $S'$ consists of $(0,0)$ pairs. In each step of the algorithm one constructs the arcs of a digraph with vertices $v_{1},\dots ,v_{n}$, i.e. if it is possible to reduce the list $S$ to $S'$, then we add arcs $(v_{i},v_{1}),(v_{i},v_{2}),\dots ,(v_{i},v_{b_{i}-1}),(v_{i},v_{b_{i}+1})$. When the list $S$ cannot be reduced to a list $S'$ of nonnegative integer pairs in any step of this approach, the theorem proves that the list $S$ from the beginning is not digraphic.

## Kleitman–Wang algorithm (maximum choice of a pair)

The algorithm is based on the following theorem.

Let $S=((a_{1},b_{1}),\dots ,(a_{n},b_{n}))$ be a finite list of nonnegative integers such that $a_{1}\geq a_{2}\geq \cdots \geq a_{n}$ and let $(a_{i},b_{i})$ be a pair such that $(b_{i},a_{i})$ is maximal with respect to the lexicographical order under all pairs $(b_{1},a_{1}),\dots ,(b_{n},a_{n})$. List $S$ is digraphic if and only if the finite list $S'=((a_{1}-1,b_{1}),\cdots ,(a_{b_{i}-1}-1,b_{b_{i}-1}),(a_{b_{i}},0),(a_{b_{i}+1},b_{b_{i}+1}),(a_{b_{i}+2},b_{b_{i}+2}),\dots ,(a_{n},b_{n}))$ has nonnegative integer pairs and is digraphic.

Note that the list $S$ must not be in lexicographical order as in the first version. If the given list $S$ is digraphic, then the theorem will be applied at most $n$ times, setting in each further step $S:=S'$. This process ends when the whole list $S'$ consists of $(0,0)$ pairs. In each step of the algorithm, one constructs the arcs of a digraph with vertices $v_{1},\dots ,v_{n}$, i.e. if it is possible to reduce the list $S$ to $S'$, then one adds arcs $(v_{i},v_{1}),(v_{i},v_{2}),\dots ,(v_{i},v_{b_{i}-1}),(v_{i},v_{b_{i}+1})$. When the list $S$ cannot be reduced to a list $S'$ of nonnegative integer pairs in any step of this approach, the theorem proves that the list $S$ from the beginning is not digraphic.

