Almost disjoint spanning trees: relaxing the conditions for completely independent spanning trees

Benoit Darties (1) , Nicolas Gastineau (2) , Olivier Togni (1)

1 Le2i - Laboratoire Electronique, Informatique et Image [UMR6306]
2 LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision

The search of spanning trees with interesting disjunction properties has led to the introduction of edge-disjoint spanning trees, independent spanning trees and more recently completely independent spanning trees. We group together these notions by defining (i, j)-disjoint spanning trees, where i (j, respectively) is the number of vertices (edges, respectively) that are shared by more than one tree. We illustrate how (i, j)-disjoint spanning trees provide some nuances between the existence of disjoint connected dominating sets and completely independent spanning trees. We prove that determining if there exist two (i, j)-disjoint spanning trees in a graph G is NP-complete, for every two positive integers i and j. Moreover we prove that for square of graphs, k-connected interval graphs, complete graphs and several grids, there exist (i, j)-disjoint spanning trees for interesting values of i and j.

Domaines: Mathématique discrète [cs.DM]

Benoit Darties, Nicolas Gastineau, Olivier Togni. Almost disjoint spanning trees: relaxing the conditions for completely independent spanning trees. 2017. ⟨hal-01476364⟩