Packing in trees

Abstract

Let G be a graph and let v be a vertex of G. The open neighbourhood N( v) of v is the set of all vertices adjacent with v in G, while the closed neighbourhood of v is N( v) ∪ ( v). A packing of a graph G is a set of vertices whose closed neighbourhoods are pairwise disjoint. Equivalently, a packing of a graph G is a set of vertices whose elements are pairwise at distance at least 3 apart in G. The lower packing number of G, denoted ϱ L( G), is the minimum cardinality of a maximal packing of G while the ( upper) packing number of G, denoted ϱ( G), is the maximum cardinality among all packings of G. An open packing of G is a set of vertices whose open neighbourhoods are pairwise disjoint. The lower open packing number of G, denoted ϱ L 0( G), is the minimum cardinality of a maximal open packing of G while the (upper) open packing number of G, denoted ϱ 0( G), is the maximum cardinality among all open packings of G. We present upper bounds on the packing number and the lower packing number of a tree. Bounds relating the packing numbers and open packing numbers of a tree are established.