An end-faithful spanning tree counterexample

An end-faithful spanning tree counterexample

We find an infinitely-connected graph in which every spanning tree has a 2-way infinite path. This disproves Halin's well-known "end-faithful spanning tree" conjecture and also disproves a recent conjecture of Siráñ.

On End-faithful Spanning Trees in Infinite Graphs

Let G be an infinite connected graph. A ray (from v) in G is a 1-way infinite path in G (with initial vertex v). An infinite connected subgraph of a ray R ⊂ G is called a tail of R. If X ⊂ G is finite, the infinite component of R\X will be called the tail of R in G\X. The following assertions are equivalent for rays P,Q ⊂ G: (i) There exists a ray R ⊂ G which meets each of P and Q infinitely of...

An optimal minimum spanning tree algorithm

This thesis describes the optimal minimum spanning tree algorithm given by Pettie and Ramachandran (in Journal of the ACM, 2002). The algorithm presented finds a minimum spanning tree of a graph with n vertices and m edges deterministically in time O (T ∗ (m,n)), where T ∗ is the minimum number of edge-weight comparisons needed to determine the solution of the problem. The function T ∗ is the d...

An Amortized Search Tree Analysis for k-Leaf Spanning Tree

Figure 12.2: A Spanning Tree Produced by an On-Line Small Spanning Tree Algorithm

In all of the previous chapters, algorithms are designed under the assumption that the whole picture of the data are available for us before the algorithm is executed. That is, problems are solved together with complete information of the data. However, this is not necessarily true in reality. Consider the disk scheduling problem. The requests of disk servers are totally unknown to the algorith...