For the Traveling Salesman Problem (TSP) on Halin graphs with three types of cost functions: sum, bottleneck and balanced and with arbitrary real edge costs we compute in polynomial time the persistency partition EAll, ESome, ENone of the edge set E, where: EAll = {e ∈ E, e belongs to all optimum solutions}, ENone = {e ∈ E, e does not belong to any optimum solution} and ESome = {e ∈ E, e belong...

a convincing exposition of his abnormalities. But although it gives an excellent psychological case-study, the bulk of it is written in terms which will also hold the interest of the layman. Its style is lucid, light and enlivened by flashes of humour. It is illustrated by several excellent photographs which themselves throw much light on Hess:s mentality. Apart from psychopathology, most inter...

Nash-Williams proved that the edges of a k-edge connected (undirected) graph can be oriented such that the resulting directed graph is ⌊ 2 ⌋-edge connected. A long-standing goal in the area is to obtain analogous results for other types of connectivity, such as node connectivity, element connectivity, and hypergraph edge connectivity. We focus on two special classes of graphs, namely, incidence...

An outerplanar (also called circular, convex, one-page) drawing of an n-vertex graph G is a drawing in which the vertices are placed on a circle and each edge is drawn using one straight-line segment. We derive exact results for the minimal number of crossings in any outerplanar drawings of the following classes of graphs: 3-row meshes, Halin graphs and complete p−partite graphs with equal size...

We prove that if a rayless tree T is mutually embeddable and non-isomorphic with another rayless tree, then T is mutually embeddable and non-isomorphic with infinitely many rayless trees. The proof relies on a fixed element theorem of Halin, which states that every rayless tree has either a vertex or an edge that is fixed by every self-embedding. We state a conjecture that proposes an extension...