Given a graph G = (V; E), k distinct vertices u 1 ; u 2 ; 1 1 1, u k 2 V and k natural numbers n 1 ; n 2 ; 1 1 1 ; n k such that P k i=1 n i = jV j, we wish to find a partition V 1 ; V 2 ; 1 1 1 ; V k of the vertex set V such that u i 2 V i , jV i j = n i , and V i induces a connected subgraph of G for each i, 1 i k. Such a partition is called a k- partition of G. The problem of finding a k-partition of a general graph is NP-hard [DF85], and hence it is very unlikely that there is a polynomial-time algorithm to solve the problem. Although not every graph has a k-partition, Gyori and Lov'asz independently proved that every k-connected graph has a k-partition for any u 1 ; u 2 ; 1 1 1 ; u k and n 1 ; n 2 ; 1 1 1 ; n k [G78, L77]. However, their proofs do not yield any polynomial-time algorithm for actually finding a k- ...

x 1 Introduction 1 1.1 Historical Background . . .............................. 4 1.2 Drawing Styles . ................................... 4 1.2.1 Polyline drawings .............................. 5 1.2.2 Planar drawings ............................... 5 1.2.3 Straight line drawings ............................ 6 1.2.4 Orthogonal drawings . . ........................... 7 1.2.5 Grid drawings ................................ 8 1.3 Properties of Drawings ................................ 9 1.4 Scope of this Thesis .................................. 10 1.4.1 Rectangular drawings . . . ......................... 11 1.4.2 Orthogonal drawings . . ........................... 12 1.4.3 Box-rectangular drawings ........................... 14 1.4.4 Convex drawings . . ............................. 16 1.5 Summary ....................................... 16 2 Preliminaries 20 2.1 Basic Terminology .................................. 20 2.1.1 Graphs and Multigraphs ........................... 20 i CO...

Consider a communication network or an undirected graph G in which a limited number of link and/or node faults F might occur. A routing ae for the network(at most one path called route for each ordered pair of nodes) mustbechosen without knowing which components might be faulty. The diameter of the surviving route graph R(G# ae)=F , where two nonfaulty nodes x and y are connected by a directed edge if there are no faults on the route from x to y, could be one of the fault-tolerant measures for the routing ae. In this paper, wesurvey the results about the diameter of the surviving route graph for general graphs. Keywords: Fault tolerance, fixed routing, k-connected graph, diameter, distributed computing 1