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

Abstract : 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 dening (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.
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Discrete Applied Mathematics, Elsevier, 2018, 236, pp.124-136. 〈10.1016/j.dam.2017.11.018〉
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Benoit Darties, Nicolas Gastineau, Olivier Togni. Almost disjoint spanning trees: Relaxing the conditions for completely independent spanning trees. Discrete Applied Mathematics, Elsevier, 2018, 236, pp.124-136. 〈10.1016/j.dam.2017.11.018〉. 〈hal-01715892〉

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