A Note on the Complexity of Directed Clique

For a directed graph $G$, and a linear order $\ll$ on the vertices of $G$, we define backedge graph $G^\ll$ to be the undirected graph on the same vertex set with edge ${u,w}$ in $G^\ll$ if and only if $(u,w)$ is an arc in $G$ and $w \ll u$. The directed clique number of a directed graph $G$ is defined as the minimum size of the maximum clique in the backedge graph $G^\ll$ taken over all linear orders $\ll$ on the vertices of $G$. A natural computational problem is to decide for a given directed graph $G$ and a positive integer $t$, if the directed clique number of $G$ is at most $t$. This problem has polynomial algorithm for $t=1$ and is known to be \NP-complete for every fixed $t\ge3$, even for tournaments. In this note we prove that this problem is $Σ^\mathsf{P}_{2}$-complete when $t$ is given on the input.

💡 Research Summary

This paper investigates the computational complexity of a graph‑theoretic parameter defined for directed graphs, called the directed clique number (denoted →ω(G)). For a directed graph G and a linear order ≪ on its vertices, the back‑edge set BE(G,≪) consists of all unordered pairs {u,w} such that (u,w) is an arc of G and w ≪ u. The undirected back‑edge graph G^≪ is formed from V(G) and the edge set BE(G,≪). The size of a maximum clique in G^≪ is ω(G^≪). The directed clique number →ω(G) is the minimum possible ω(G^≪) over all linear orders ≪ on V(G). Intuitively, →ω(G) measures how “close” a directed graph can be made to a transitive orientation by reordering its vertices.

The natural decision problem is: given a directed graph G and an integer t, decide whether →ω(G) ≤ t. When t is fixed, the problem is trivial for t=1 (transitivity can be checked in linear time) and NP‑complete for every fixed t≥3, even on tournaments. The open question is the complexity when t is part of the input. The authors prove that this general problem is Σ₂^P‑complete, i.e., complete for the second level of the polynomial hierarchy.

The proof proceeds by a reduction from a known Σ₂^P‑complete problem, called Existential‑2‑Level‑3‑CNF. An instance of this problem consists of a 3‑CNF formula φ(x₁,…,x_a, y₁,…,y_b) where the variables are partitioned into an existential block X and a universal block Y. The question is whether ∃X ¬∃Y φ holds. This problem is equivalent to the canonical Σ₂^P‑complete problem “∃X ∀Y φ” and can be obtained from Existential‑2‑Level‑SAT by a simple transformation.

The reduction constructs, from a given formula, a directed graph G whose structure forces any linear order ≪ that avoids a large clique in the back‑edge graph G^≪ to encode a satisfying assignment for the X‑variables that falsifies every clause under any Y‑assignment. The construction uses three families of gadgets:

1. Binary Gadget – For each occurrence of an X‑variable x_i in a clause C_k, a subgraph G(k)_x_i is built. It contains three complete directed subgraphs A′ (size t−2), A_F (size t−1) and A_T (size t−1), together with four special vertices: x(k)_i, x(k)_i^F, x(k)_i^T, and w(k)i. The arcs are arranged so that, under any order ≪ that keeps G^≪ K{2c}‑free (where c is the number of clauses), exactly one of the back‑edges {x(k)_i, x(k)_i^F} or {x(k)_i, x(k)_i^T} appears. This choice is interpreted as setting x_i to false or true, respectively, for that particular occurrence.

2. Copy Gadget – When the same X‑variable appears in two different clauses C_k and C_ℓ, a copy gadget G(k,ℓ)_x_i links the two corresponding binary gadgets. It adds four additional complete directed subgraphs A₁,…,A₄ (size t−3) and a middle layer that is a copy of the binary gadget. The crucial property is that the presence of the edge {x(k)_i, x(k)_i^F} in the back‑edge graph forces the same edge {x(ℓ)_i, x(ℓ)_i^F} to appear, and similarly for the “true” edges. Thus the assignment chosen for x_i is consistent across all its occurrences.

3. Clause Gadget – For each clause C_k, six vertices are introduced, grouped in three pairs. For a literal that is a positive occurrence of an X‑variable, the pair consists of x(k)_i and x(k)_i^T; for a negative occurrence, it consists of x(k)_i and x(k)_i^F. For a Y‑literal (positive or negative) the pair consists of two vertices y(k)_j^A and y(k)j^B connected by opposite arcs. Between any two groups belonging to different clauses, if the corresponding literals involve different variables or the same variable with the same sign, all possible eight directed arcs are added. This dense inter‑group connectivity ensures that, if a clause is not satisfied by the chosen X‑assignment, the corresponding six vertices together with the appropriate auxiliary vertices will form a K{2c} in the back‑edge graph.

The parameter t is set to 2c−1, where c is the number of clauses. This choice guarantees that any K_{2c} in a back‑edge graph must involve vertices from at least one of the gadget’s internal complete subgraphs, which forces the order ≪ to respect the gadget’s intended behavior. The reduction runs in polynomial time because the number of vertices and arcs added is polynomial in the size of the original formula.

The correctness argument has two directions:

If the formula is a YES instance (i.e., there exists an assignment ν to X such that for every Y‑assignment the formula is false), the authors construct an order ≪ as follows. Within each binary gadget, the order of the four special vertices reflects ν_i (true → x(k)_i ≪ x(k)_i^F ≪ w(k)_i ≪ x(k)_i^T; false → x(k)_i^F ≪ w(k)i ≪ x(k)i^T ≪ x(k)i). The internal vertices of the complete subgraphs can be placed arbitrarily. Because of the choice of t, the back‑edge graph induced by any gadget is K{2c}‑free. The copy gadgets preserve consistency, and the clause gadgets cannot create a K{2c} because each clause contains at least one literal satisfied by ν, which blocks the formation of the forbidden clique. Hence G^≪ contains no K{2c}, implying →ω(G) ≤ 2c−1.

If the formula is a NO instance, any order ≪ that avoids a K_{2c} would induce a consistent assignment to the X‑variables (by the binary and copy gadgets). Since the formula is false, there must be a clause all of whose literals evaluate to false under this assignment. The clause gadget for that clause, together with the dense inter‑group arcs, would then inevitably produce a K_{2c} in the back‑edge graph, contradicting the assumption. Therefore, no order can keep G^≪ K_{2c}‑free, and →ω(G) > 2c−1.

Thus the reduction establishes Σ₂^P‑hardness. Since the problem is clearly in Σ₂^P (it can be expressed as ∃≪ ∀A⊆V,|A|=t+1: A is not a clique in G^≪), the decision problem “DirectedClique” is Σ₂^P‑complete.

Beyond the main theorem, the paper discusses the related problem TournamentClique, where the input graph is required to be a tournament. The authors conjecture that TournamentClique is also Σ₂^P‑complete, but note that constructing tournaments with high directed clique numbers is challenging. They cite known lower bounds (Ω(log n) for random tournaments) and provide a trivial upper bound →ω(T) ≤ p·2^{|V(T)|}. They also observe that the complete directed graph Q_n has directed clique number n, showing that high values are achievable in general digraphs.

Finally, the paper mentions potential future work, including explicit constructions of tournaments with polynomially large directed clique numbers, and exploring the relationship between directed clique number and other digraph width parameters such as dichromatic number.

In summary, the authors settle the long‑standing open question about the complexity of the directed clique number when the threshold t is part of the input: the problem sits precisely at the second level of the polynomial hierarchy, being Σ₂^P‑complete. This result deepens our understanding of how ordering‑based parameters behave computationally and opens new avenues for studying similar parameters on directed graphs and tournaments.