Kleitman–Wang algorithms 📖 Wikipedia

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 ^[1] gave these algorithms in 1973.

The algorithm is based on the following theorem.

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

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

The algorithm is based on the following theorem.

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

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

• Fulkerson–Chen–Anstee theorem

• Kleitman, D. J.; Wang, D. L. (1973), "Algorithms for constructing graphs and digraphs with given valences and factors", Discrete Mathematics, 6: 79–88, doi:10.1016/0012-365x(73)90037-x