Finding forest-orderings of tournaments is NP-complete

TL;DR

This work establishes that the Feedback Arc Set problem for forest-structured backedge graphs in tournaments is NP-complete by a careful reduction from 3-SAT. The authors construct a polynomial-size tournament $T_{\mathcal{I}}$ from a 3-SAT instance $\mathcal{I}$, combining a base tournament with variable/literal gadgets and a network of back-arc matchings that encode the SAT constraints; a forest-ordering of $T_{\mathcal{I}}$ corresponds precisely to a satisfying assignment of $\mathcal{I}$. They demonstrate both directions: a forest-ordering yields a valid assignment, and a satisfying assignment yields a forest-ordering, using a detailed degeneracy-peeling argument. The result links structural tournament parameters (dichromatic number, clique number) to SAT-complexity and highlights the role of forest-orderings in understanding tournament structure, while also noting open questions about related graph classes and the practical implications for approximating or recognizing such orderings.

Abstract

Given a class of (undirected) graphs $\mathcal{C}$, we say that a Feedback Arc Set (FAS for short) $F$ is a $\mathcal{C}$-FAS if the graph induced by the edges of $F$ (forgetting their orientations) belongs to $\mathcal{C}$. We show that deciding if a tournament has a $\mathcal{C}$-FAS is NP-complete when $\mathcal{C}$ is the class of all forests. We are motivated by connections between $\mathcal{C}$-FAS and structural parameters of tournaments, such as the dichromatic number, the clique number of tournaments, and the strong Erdős-Hajnal property.

Figures (10)

• Figure 1: A tournament on eight vertices admitting a unique forest-ordering (left to right on the figure). Forward arcs are faded.
• Figure 2: A back-arc matching from a transitive tournament $T_2$ to a transitive tournament $T_1$, both on three vertices, and whose topological orderings are left-to-right. The thick arcs are precisely the arcs of $T^{\prec}$, and they form a matching. Arcs inside $T_1$ and $T_2$ are omitted.
• Figure 3: The figure represents a vertex $v \in \mathcal{V}$, $M_v$ and $G_v = N_v \sqcup Y_v$ ordered as in $\prec^*$. Forward-arcs are note drawn. The tournament induced by $M_x$ is the same as the one depicted in \ref{['figure:magical-tournament']}, and thus the back-arcs in $T[M_v]$ induce a tree. A thick arc represents all arcs in that orientation.
• Figure 4: The figure represents a vertex $z \in \mathcal{L}$, $M_z$ and $G_z = N_z \sqcup Y_z$ ordered as in $\prec^*$. Forward-arcs are not drawn. The tournament induced by $M_z$ is the same as the one depicted in \ref{['figure:magical-tournament']}, and thus the back-arcs in $T[M_z]$ induce a tree. A thick arc represents all arcs in that orientation.
• Figure 5: Back-arc matchings from $N^{v}_{\overline{v}}$ to $N^{\overline{v}}_v$, where $v,\overline{v} \in \mathcal{V}$, and from $N^{\overline{v}}_{\ell}$ and $N^{j}_{\overline{v}}$, where $\ell$ is the $j^{th}$ occurrence of $\overline{x}$ and $x$ is the variable associated with the pair $v,\overline{v}$. Vertices are ordered as in $\prec^*$. Non-drawn arcs are all forward-arcs.

Theorems & Definitions (15)

• theorem 1
• lemma 2
• proof
• lemma 3
• lemma 4
• proof
• lemma 5
• proof
• lemma 6
• proof