Abstract Traditionally, the trees studied in infinite graphs are of height at most ω , with each node adjacent to its parent and children (and every branch tree inducing a path or ray). However, there is also method, systematically introduced by Brochet Diestel, turning arbitrary well-founded order T into graphs, way such that -branch induces generalised sense Rado. This article contains an int...

The concepts of simplicial decompositions, tree-decompositions and simplicial tree-decompositions were all inspired by a common forerunner: the decompositions of finite graphs used by K. Wagner in his classic paper [ 13 ], in which he proved the equivalence of the 4-Colour-Conjecture to Hadwiger’s Conjecture for n = 5. To show that the 4CC implies Hadwiger’s Conjecture (for n = 5), Wagner used ...

If two rays P,Q ⊂ G satisfy (i)–(iii), we call them end-equivalent in G. An end of G is an equivalence class under this relation, and E(G) denotes the set of ends of G. For example, the 2-way infinite ladder has two ends, the infinite grid Z×Z and every infinite complete graph have one end, and the dyadic tree has 20 ends. This paper is concerned with the relationship between the ends of a conn...

A piecewise linear simple curve in the plane made up of k+1 line segments, each of which is either horizontal or vertical, with consecutive segments being of different orientation is called a k-bend path. Given a graph G, a collection of k-bend paths in which each path corresponds to a vertex in G and two paths have a common point if and only if the vertices corresponding to them are adjacent i...

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). A graph G is called fully subdivided if it is obtained from another graph H by replacing every edge by a path of length at least two. Fully subdi...

Given a graph G with n vertices and a set S of n points in the plane, a point-set embedding of G on S is a planar drawing such that each vertex of G is mapped to a distinct point of S. A straight-line point-set embedding is a point-set embedding with no edge bends or curves. The point-set embeddability problem is NP-complete, even when G is 2-connected and 2-outerplanar. It has been solved poly...

A side skirt is a planar rooted tree T, T≠P2, where the root of T vertex degree at least two, and all other vertices except leaves are three. reduced Halin graph or skirted plane G=T∪P, skirt, P path connecting in order determined by embedding T. The structure graphs contains both symmetry asymmetry. For n≥2 Pn=v1v2v3⋯vn as length n−1, we call Cartesian product G Pn, n-generalized prism over G....

This thesis is based upon the premise that access to health care is a fundamental human right and should never be dictated by profit, class, gender, or national identity. Given the quixotism of this objective for universal health care coverage, I am studying the works of the most prominent proponents of socialized healthcare—Rudolf Virchow, Henry E. Sigerist, and Paul Farmer. According to the W...