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

Paper

Abstract

The {\it inversion} of a set

of vertices in a digraph

consists of reversing the direction of all arcs of

. We study

(resp.

) which is the minimum number of inversions needed to transform

into a

-arc-strong (resp.

-strong) digraph and

. We show :

;

for any fixed positive integers

and

, deciding whether a given oriented graph

with

satisfies

is NP-complete;

for any fixed positive integers

and

, deciding whether a given oriented graph

with

satisfies

is NP-complete;

is a tournament of order at least

, then

, and

;

for some tournament

of order

;

is a tournament of order at least

(resp.

), then

(resp.

);

for every

, there exists

such that

for every tournament

on at least

vertices.