Making an oriented graph acyclic using inversions of bounded or prescribed size

TL;DR

This work analyzes rendering an oriented graph acyclic via inversions of vertex subsets, focusing on inversions of prescribed size (=p) and bounded size (≤p). It reveals a sharp complexity split: (n−1)-Invertibility is NP-complete, while (=p)Invertibility is polynomial-time decidable for all other p; it fully characterizes (=p)-invertibility for tournaments across p mod 4 and extends a polynomial-time test to general digraphs. The authors derive asymptotic bounds on the corresponding inversion numbers, establish tight relations with the feedback arc set in the even-p case, and show fundamental separations for odd p. Finally, they address parameterized complexity, proving NP-hardness and W[1]-hardness results, and provide kernelizations (O(k^2 p^3)) for tournament variants, highlighting practical tractability when p and k are small.

Abstract

Given an oriented graph $D$, the inversion of a subset $X$ of vertices consists in reversing the orientation of all arcs with both endpoints in $X$. When the subset $X$ is of size $p$ (resp. at most $p$), this operation is called an $(=p)$-inversion (resp. $(\leq p)$-inversion). Then, an oriented graph is $(=p)$-invertible if it can be made acyclic by a sequence of $p$-inversions. We observe that, for $n=|V(D)|$, deciding whether $D$ is $(=n-1)$-invertible is equivalent to deciding whether $D$ is acyclically pushable, and thus NP-complete. In all other cases, when $p \neq n-1$, we construct a polynomial-time algorithm to decide $(=p)$-invertibility. We then consider the $(= p)$-inversion number, $\text{inv}^{= p}(D)$ (resp. $(\leq p)$-inversion number, $\text{inv}^{\leq p}(D)$), defined as the minimum number of $(=p)$-inversions (resp. $(\leq p)$-inversions) rendering $D$ acyclic. We show that every $(=p)$-invertible digraph $D$ satisfies $\text{inv}^{= p}(D) \leq |A(D)|$ for every integer $p\geq 2$. When $p$ is even, we bound $\text{inv}^{= p}$ by a (linear) function of the feedback arc set number, and rule out the existence of any bounding function for odd $p$. Finally, we study the complexity of deciding whether the $(= p)$-inversion number, or the $(\leq p)$-inversion number, of a given oriented graph is at most a given integer $k$. For any fixed positive integer $p \geq 2$, when $k$ is part of the input, we show that both problems are NP-hard even in tournaments. In general oriented graphs, we prove $W[1]$-hardness for both problems when parameterized by $p$, even for $k=1$. In contrast, we exhibit polynomial kernels in $p + k$ for both problems in tournaments.

Figures (1)

• Figure 1: An illustration of an instance $(G,(V_1,V_2,V_3))$ of MCC and the corresponding oriented graph $D$. The vertices of $V_1,V_2$, and $V_3$ are marked in blue, red, and green in $G$, respectively. For $i \in [3]$, the directed cycle $C_i$ in $D$ is marked in the same colour as $V_i$ in $G$. All vertices of $Z$ and arcs of $D$ incident to $Z$ are marked in black. The names of the vertices in $Z$ have been omitted due to space restrictions.

Theorems & Definitions (92)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.7
• Corollary 1.7
• Theorem 1.8