The problem A graph G consists of a set V G of vertices and a set E G of edges each of which is an ordered pair of vertices These objects despite their simple struc ture can be used to model important properties of a wide variety of math ematical and physical systems One of their most important applications is to the study of routing in networks Here the vertices represent sites cities computer...

We propose a new arithmetic for non-empty rooted unordered trees simply called trees. After discussing tree representation and enumeration, we define the operations of tree addition, multiplication, and stretch, prove their properties, and show that all trees can be generated from a starting tree of one vertex. We then show how a given tree can be obtained as the sum or product of two trees, th...

The graph G contains a graph H as a minor if there exist pair-wise disjoint sets {Si ⊆ V (G)|i = 1, . . . , |V (H)|} such that for every i, G[Si] is a connected subgraph and for every edge uv in H, there exists an edge of G with one end in Su and the other end in Sv. A rooted H minor in G is a minor where each Si of minor contains a predetermined xi ∈ V (G). We prove that if the constant c is s...

The problem of computing a minimum cost subgraph D′ = (V,A′) of a directed graph D = (V,A) so that D′ contains k edge-disjoint paths from a specified root r ∈ V to every other node in V is known to be nicely solvable since it can be formulated as a matroid intersection problem. A corresponding problem when openly disjoint paths are requested rather than edge-disjoint was solved in [12] with the...