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The search of spanning trees with interesting disjunction properties has ledto the introduction of edge-disjoint spanning trees, independent spanning treesand more recently completely independent spanning trees. We group togetherthese notions by defining (i, j)-disjoint spanning trees, where i (j,respectively) is the number of vertices (edges, respectively) that are sharedby more than one tree. We illustrate how (i, j)-disjoint spanning trees providesome nuances between the existence of disjoint connected dominating sets andcompletely independent spanning trees. We prove that determining if there existtwo (i, j)-disjoint spanning trees in a graph G is NP-complete, for every twopositive 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.

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