Induced tree covering and the generalized Yutsis property

The Yutsis property of a graph G is the property of partitioning its vertex set into two induced trees. Although recognizing Yutsis graphs is NP-complete even on planar graphs, it is still possible to consider two even more challenging problems: (i) recognizing k-Yutsis graphs, which are graphs that have their vertex sets partitioned into k induced trees, for a fixed $k\ge 2$; (ii) determining the tree cover number of a given graph G, i.e., the minimum number of vertex-disjoint induced trees covering all vertices of G. We prove that determining the tree cover number of a split graph G is NP-hard, contrasting with the polynomial-time recognition of k-Yutsis chordal graphs. We also investigate the tree cover number computation and the k-Yutsis graph recognition concerning treewidth and clique-width parameterizations.