Necessary and Sufficient Condition for Existence of Locally-balanced 2-partition of a Tree under the Extended Definition of a Neighbourhood of a Vertex - Balikyan , Suren V.; Kamalian, Rafayel R. (Mathematical Problems of Computer Science, 2021)
A necessary and sufficient condition is obtained for the problem of partitioning of the set of vertices of a tree G into two disjoint sets V1 and V2 such that it satisfies the condition ||λ(v \ V1|–|λ( v) \ V2|| ≤ 1 for any vertex v of G, where λ(v) is the set of all vertices of G the distance of which from v does not exceed 1.