DeepAI

# Invertibility of digraphs and tournaments

For an oriented graph D and a set X⊆ V(D), the inversion of X in D is the digraph obtained by reversing the orientations of the edges of D with both endpoints in X. The inversion number of D, inv(D), is the minimum number of inversions which can be applied in turn to D to produce an acyclic digraph. Answering a recent question of Bang-Jensen, da Silva, and Havet, we show that for each fixed k∈ℕ the problem of deciding whether a tournament T has inv(T)≤ k is solvable in time O(| V(T)| ^2). This exponent is optimal for all k. On the other hand, we build on their work to prove their conjecture that for k≥ 1 the problem of deciding whether a general oriented graph D has inv(D)≤ k is NP-complete. We also construct oriented graphs with inversion number equal to twice their cycle transversal number, confirming another conjecture of Bang-Jensen, da Silva, and Havet, and we provide a counterexample to their conjecture concerning the inversion number of so-called 'dijoin' digraphs while proving that it holds in certain cases. Finally, we asymptotically solve the natural extremal question in this setting, improving on previous bounds of Belkhechine, Bouaziz, Boudabbous, and Pouzet to show that the maximum inversion number of an n-vertex tournament is (1+o(1))n.

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05/10/2021

### On the inversion number of oriented graphs

Let D be an oriented graph. The inversion of a set X of vertices in D co...
12/18/2022

### Problems, proofs, and disproofs on the inversion number

The inversion of a set X of vertices in a digraph D consists in reversin...
09/13/2022

### Proving a directed analogue of the Gyárfás-Sumner conjecture for orientations of P_4

An oriented graph is a digraph that does not contain a directed cycle of...
03/21/2023

### On the minimum number of inversions to make a digraph k-(arc-)strong

The inversion of a set X of vertices in a digraph D consists in reversin...
08/23/2022

### Homomorphically Full Oriented Graphs

Homomorphically full graphs are those for which every homomorphic image ...
01/24/2023

### Restricted optimal pebbling is NP-hard

Consider a distribution of pebbles on a graph. A pebbling move removes t...
05/16/2018

### The Crossing Number of Single-Pair-Seq-Shellable Drawings of Complete Graphs

The Harary-Hill Conjecture states that for n≥ 3 every drawing of K_n has...