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This paper gives a purely set-theoretical extension theorem on iterative roots, which generalizes the results about PM functions by Zhang and his collaborators. Applying this extension theorem, we construct all iterative roots with infinitely many discontinuities for a class of Markov maps. [ABSTRACT FROM AUTHOR]

One famous open problem in graph theory is the Graceful Tree Conjecture, which states that every finite tree has a graceful labeling. In 1973, Kotzig (Util Math 4:261-290, ) proved that if a leaf of a long enough path is identified with any vertex of an arbitrary tree, the resulting tree is graceful. In this paper, we prove that if the center of a large enough star is identified with any vertex of an arbitrary tree, the resulting tree is graceful, and we also provide an upper bound for the size of the star. [ABSTRACT FROM AUTHOR]