Digraph Realization Problem (Graph Realization Problems)

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Given a sequence $S := (a_1, b_1), \ldots, (a_n, b_n)$ with $a_i, b_i \in \mathbb{Z}_0^+$, does there exist a directed graph (no parallel arcs allowed) with labeled vertex set $V := \{v_1, \ldots , v_n\}$ such that for all $v_i \in V$ indegree and outdegree of $v_i$ match exactly the given numbers $a_i$ and $b_i$, respectively?

Related Problems

Subproblem: DAG Realization Problem


$n$: number of degree pairs

Table of Algorithms

Name Year Time Space Approximation Factor Model Reference
Kleitman–Wang Algorithm 1973 $O(n)$ $O(n)$ Exact Deterministic Time
Fulkerson–Chen–Anstee 1982 $O(n)$ $O({1})$ Exact Deterministic Time

Time Complexity Graph

Graph Realization Problems - Digraph Realization Problem - Time.png