Non-separating spanning trees and out-branchings in digraphsof independence number 2

TL;DR

This paper investigates the existence of non-separating spanning trees and out-branchings in digraphs with independence number 2, establishing conditions on connectivity and minimum in-degree for their existence.

Contribution

It provides new sufficient conditions for non-separating out-branchings and spanning trees in digraphs with independence number 2, including minimum in-degree thresholds.

Findings

Every 2-arc-strong digraph with independence number 2 and min in-degree ≥5 has a non-separating out-branching.

Every 2-arc-strong oriented graph with independence number 2 and min in-degree ≥3 has a non-separating out-branching.

For digraphs with independence number ≤2 and at least 14 vertices, a non-separating spanning tree exists.

Abstract

A subgraph H= (V, F) of a graph G= (V,E) is non-separating if G-F, that is, the graph obtained from G by deleting the edges in F, is connected. Analogously we say that a subdigraph X= (V,B) of a digraph D= (V,A) is non-separating if D-B is strongly connected. We study non-separating spanning trees and out-branchings in digraphs of independence number 2. Our main results are that every 2-arc-strong digraph D of independence number alpha(D) = 2 and minimum in-degree at least 5 and every 2-arc-strong oriented graph with alpha(D) = 2 and minimum in-degree at least 3 has a non-separating out-branching and minimum in-degree 2 is not enough. We also prove a number of other results, including that every 2-arc-strong digraph D with alpha(D)<=2 and at least 14 vertices has a non-separating spanning tree and that every graph G with delta(G)>=4 and alpha(G) = 2 has a non-separating hamiltonian path.