In a paper from 2001 (Journal of the LMS), Diestel and Leader offered proof that connected graph has normal spanning tree if only it no minor obtained canonically either an ( ℵ 0 , 1 ) -regular bipartite or order-theoretic Aronszajn tree. particular, this refuted earlier conjecture Halin's first these obstructions was needed to characterize graphs with trees. However, Leader's contains gap, the...

In this paper, we present algorithms for enumeration of spanning trees in undi-rected graphs, with and without weights. The algorithms use a search tree technique to construct a computation tree. The computation tree can be used to output all spanning trees by outputting only relative changes between spanning trees rather than the entire spanning trees themselves. Both the construction of the c...

The minimum spanning trees problem is to find k edge-disjoint spanning trees in a given undirected weighted graph. It can be solved in polynomial time. In the k minimum spanning trees problem with diameter bounded below (k-MSTBB) there is an additional requirement: a diameter of every spanning tree must be not less than some predefined value d. The k-MSTBB is NP hard. We propose an asymptotical...

Given a graph G, a set of spanning trees of G are completely independent if for any vertices x and y, the paths connecting them on these trees have neither vertex nor edge in common, except x and y. In this paper, we prove that for graphs of order n, with n ≥ 6, if the minimum degree is at least n−2, then there are at least n/3 completely independent spanning trees. Keyword: Completely independ...

Spanning trees have been studied extensively and have many practical applications. From astronomers using spanning trees to find quasar superstructures, to biomedical engineers detecting actin fibers in cell images tree enumeration remains an interesting problem. In this paper we examine a simple, practical, algebraic approach for the enumeration of spanning trees in complete and non-complete g...

Given a graph G, a designated vertex r and a natural number k, we wish to find k “independent” spanning trees of G rooted at r, that is, k spanning trees such that, for any vertex v, the k paths connecting r and v in the k trees are internally disjoint in G. In this paper we give a linear-time algorithm to find four independent spanning trees in a 4-connected planar graph rooted at any vertex.

Deo and Micikevicius recently gave a new bijection for spanning trees of complete bipartite graphs. In this paper we devise a generalization of Deo and Micikevicius’s method, which is also a modification of Olah’s method for encoding the spanning trees of any complete multipartite graph K(n1, . . . , nr). We also give a bijection between the spanning trees of a planar graph and those of any of ...

Let G be an undirected graph with V vertices and E edges. We consider the problem of enumerating all spanning trees of G: In order to explicitly output all spanning trees, the output size is of (NV ), where N is the number of spanning trees. This, however, can be compressed into (N) size. In this paper, we propose a new algorithm for enumerating all spanning trees of G in such compact form. The...