On Existence of 2-partition of a Tree, Which Obeys the Given Priority - Balikyan, Suren V.; Kamalian, Rafayel R. (Mathematical Problems of Computer Science, 2021)
A necessary and sufficient condition is obtained for the problem of such partitioning of the set of vertices of a tree G into two disjoint sets V1 and V2, which, for a given function p : V (G) ! f¡1; 0; 1g with some special restriction, satisfies the condition j¸(v) \ V1j ¡ j¸(v) \ V2j = p(v) ¢ (jfvg \ V1j ¡ jfvg \ V2j) for any vertex v of G, where¸(v) is the set of all vertices of G adjacent to v.