On the $(\leq p)$-inversion diameter of oriented graphs

Abstract: In an oriented graph $\vec{G}$, the {\it inversion} of a subset $X$ of vertices consists in reversing the orientation of all arcs with both endvertices in $X$. The {\it $(\leq p)$-inversion graph} of a labelled graph $G$, denoted by ${\mathcal{I}}^{\leq p}(G)$, is the graph whose vertices are the labelled orientations of $G$ in which two labelled orientations $\vec{G}1$ and $\vec{G}_2$ of $G$ are adjacent if and only if there is a set $X$ with $|X|\leq p$ whose inversion transforms $\vec{G}_1$ into $\vec{G}_2$. In this paper, we study the {\it $(\leq p)$-inversion diameter} of a graph, denoted by $\mathrm{id}^{\leq p}(G)$, which is the diameter of its $(\leq p)$-inversion graph. We show that there exists a smallest number $Ψ_p$ with $\frac{1}{4}p - \frac{3}{2} \leq Ψ_p \leq \frac{1}{2}p^2$ such that $\mathrm{id}^{\leq p}(G) \leq \left\lceil\frac{|E(G)|}{\lfloor p/2\rfloor}\right \rceil + Ψ_p$ for all graph $G$. We then establish better upper bounds for several families of graphs and in particular trees and planar graphs. Let us denote by $\mathrm{id}^{\leq p}{\cal F}(n)$ (resp. $\mathrm{id}^{\leq p}{\cal P}(n)$) the maximum $(\leq p)$-inversion diameter of a tree (resp. planar graph) of order $n$. For trees, we show $\mathrm{id}^{\leq 3}{\cal F}(n) = \left\lceil \frac{n-1}{2}\right\rceil$, $\mathrm{id}^{\leq 4}{\cal F}(n)=\frac{3}{8}n + Θ(1)$, $\mathrm{id}^{\leq 5}{\cal F}(n)= \frac{2}{7}n + Θ(1)$, and $\mathrm{id}^{\leq p}{\cal F}(n) \leq \frac{n-1}{p- c\sqrt{p}} + 2$ with $c = \sqrt{2 + \sqrt{2}}$ for all $p\geq 6$. For planar graphs, we prove $\mathrm{id}^{\leq 3}{\cal P}(n) \leq \frac{11n}{6} - \frac{8}{3}$, $\mathrm{id}^{\leq 4}{\cal P}(n) \leq \frac{4n}{3} + \frac{10}{3}$, and $\mathrm{id}^{\leq p}{\cal P}(n) \leq \left\lceil\frac{3n-6}{\lfloor p/2\rfloor}\right \rceil + 8\lfloor p/2\rfloor - 8$ for all $p\geq 6$.

• The paper introduces the (≤p)-inversion diameter as a novel metric for assessing reconfiguration distances in oriented graphs via generalized inversion operations.
• It derives sharp upper and lower bounds for various graph classes, including trees and planar graphs, using combinatorial decomposition and edge‐coloring techniques.
• The results unify local and global transformation problems, offering precise asymptotic and exact inversion diameter values that guide future algorithmic research.

The $(\leq p)$-Inversion Diameter of Oriented Graphs

Introduction and Background

The paper investigates the reconfiguration problem on the space of labelled orientations of a graph $G$, where two orientations are connected if one can be obtained from the other via the inversion of all arcs within a subset of at most $p$ vertices. The central focus is the $(\leq p)$-inversion diameter of a graph $G$, denoted $\mathrm{id}^{(\leq p)}(G)$, which is the maximum distance between pairs of orientations in the $(\leq p)$-inversion graph $\mathcal{I}^{(\leq p)}(G)$.

While classical results consider inversion operations of size $2$ (aligning with the feedback arc set problem), this work generalizes such operations to arbitrary $p \geq 2$, thus capturing a spectrum of intermediate reconfiguration problems between local and global operations. The $G$0-inversion framework unifies several problems: for $G$1, it recovers the feedback arc set, and as $G$2 grows, it connects to total inversion diameters studied previously [Havet et al. 2024, Yuster 2025].

Formal Definitions

Given a labelled undirected graph $G$3:

• An inversion of a vertex set $G$4 transforms an orientation $G$5 into $G$6 by reversing every arc between pairs of vertices in $G$7.
• The $G$8-inversion graph $G$9 is the reconfiguration graph whose vertices are all labelled orientations of $p$0, with an edge between $p$1 and $p$2 if one can be obtained from the other by a single inversion over a set $p$3 with $p$4.
• The $p$5-inversion diameter $p$6 is the diameter of this reconfiguration graph.

Associated invariants like the $p$7-inversion number for acyclic transformation and the $p$8-converse number for transforming an orientation to its converse are also considered, with the converse number being a lower bound for the inversion diameter.

Main Theoretical Results

General Bounds

The authors establish sharp upper and lower bounds for $p$9 in terms of the graph's edge set and parameter $(\leq p)$0. The principal result is the existence of a constant $(\leq p)$1 such that

$(\leq p)$2

with bounds $(\leq p)$3, holding for every graph $(\leq p)$4. The lower bound is tight for sparse graphs such as matchings.

For dense graphs, an improved bound is given:

$(\leq p)$5

where $(\leq p)$6, matching a construction for complete bipartite graphs.

Structural Bounds for Graph Classes

The paper develops refined bounds for trees, forests, and planar graphs using combinatorial decompositions and edge-coloring techniques.

Trees

Sharp asymptotic behaviors for trees of order $(\leq p)$7 are determined for specific $(\leq p)$8:

• For $(\leq p)$9: $G$0
• For $G$1: $G$2
• For $G$3: $G$4

The general case is encapsulated as

$G$5

for a constant $G$6 and any $G$7.

Figure 1: Example of the tree $G$8 constructed to show tightness in Proposition tree4-tight for $G$9; the tree structure realizes the worst-case inversion sequence size.

Planar Graphs

For the class $\mathrm{id}^{(\leq p)}(G)$0 of planar graphs, the bounds are:

• $\mathrm{id}^{(\leq p)}(G)$1
• $\mathrm{id}^{(\leq p)}(G)$2
• For $\mathrm{id}^{(\leq p)}(G)$3: $\mathrm{id}^{(\leq p)}(G)$4

Tightness and Lower Bounds

The constructions, notably for trees (see Figure 1) and for specific planar graphs, show these upper bounds are tight up to additive constants. For $\mathrm{id}^{(\leq p)}(G)$5, Figure 1 details the explicit tree structure used for lower bound arguments.

Procedural and Combinatorial Techniques

Several combinatorial innovations underpin these bounds:

• Strong edge-colorings are used to partition edges into induced matchings, which directly control the minimum number of inversions required.
• Recursive leaf-pruning and partitioning schemes for trees achieve the aforementioned structural decompositions, minimizing the inversion sequence length.
• Planarity constraints and minor-exclusion arguments allow further improvements for planar graphs, leveraging their forbidden substructures to compress inversion sequences.

  Figure 2: Tree $\mathrm{id}^{(\leq p)}(G)$6 for $\mathrm{id}^{(\leq p)}(G)$7 as described in Proposition tree5-tight, exhibiting the extremal layout necessary to require the maximal number of $\mathrm{id}^{(\leq p)}(G)$8-inversions.

Numerical Results and Strong Claims

The strongest numerical findings include:

• For trees with $\mathrm{id}^{(\leq p)}(G)$9 vertices, the tight values for the inversion diameter for $(\leq p)$0 are given exactly.
• For planar graphs, the precise linear coefficients in $(\leq p)$1 for $(\leq p)$2 and $(\leq p)$3 inversion diameters are proven, sharpening prior bounds from the literature.
• The additive constant $(\leq p)$4 is precisely computed for small $(\leq p)$5: $(\leq p)$6 for $(\leq p)$7 and $(\leq p)$8 for $(\leq p)$9.

Implications and Future Directions

The results provide a unified framework for inversion-based reconfiguration in oriented graphs, capturing a spectrum from local to global transformations. The sharp bounds have implications for algorithmic approaches to orientation reconfiguration, especially in sparse or structured graphs.

The theoretical findings raise further open problems:

• Determination of the exact form and asymptotic behavior of $\mathcal{I}^{(\leq p)}(G)$0 for larger $\mathcal{I}^{(\leq p)}(G)$1.
• Potential to extend the methods to additional graph classes, such as minor-free or bounded treewidth graphs.
• Exploration of the computational complexity of finding shortest $\mathcal{I}^{(\leq p)}(G)$2-inversion sequences in practice, where the results give worst-case optimality guarantees.

Conclusion

This work provides a comprehensive analysis of the $\mathcal{I}^{(\leq p)}(G)$3-inversion diameter across a range of graph classes, offering tight asymptotic and, in many cases, exact values. By connecting combinatorial structure, edge partitioning techniques, and reconfiguration graph diameters, it sets the stage for further expansion into algorithmic and complexity considerations for orientation reconfiguration problems in graphs.

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Reference: "On the $\mathcal{I}^{(\leq p)}(G)$4-inversion diameter of oriented graphs" (2604.04633)