Good spanning trees in graph drawing

On Spanning 2-trees in a Graph

A k-tree is either a complete graph on k vertices or a graph T that contains a vertex whose neighbourhood in T induces a complete graph on k vertices and whose removal results in a k-tree. A subgraph of a graph is a spanning k-tree if it is a k-tree and contains every vertex of the graph. This paper is concerned with spanning 2-trees in a graph. It is shown that spanning 2-trees have close conn...

On Central Spanning Trees of a Graph

We consider the collection of all spanning trees of a graph with distance between them based on the size of the symmetric difference of their edge sets. A central spanning tree of a graph is one for which the maximal distance to all other spanning trees is minimal. We prove that the problem of constructing a central spanning tree is algorithmically difficult and leads to an NP-complete problem.

Some Appli ations of Orderly Spanning Trees inGraph Drawing ?

On relation between the Kirchhoff index and number of spanning trees of graph

Let $G=(V,E)$, $V={1,2,ldots,n}$, $E={e_1,e_2,ldots,e_m}$,be a simple connected graph, with sequence of vertex degrees$Delta =d_1geq d_2geqcdotsgeq d_n=delta >0$ and Laplacian eigenvalues$mu_1geq mu_2geqcdotsgeqmu_{n-1}>mu_n=0$. Denote by $Kf(G)=nsum_{i=1}^{n-1}frac{1}{mu_i}$ and $t=t(G)=frac 1n prod_{i=1}^{n-1} mu_i$ the Kirchhoff index and number of spanning tree...

Pitfalls of Using PQ-Trees in Automatic Graph Drawing

Since the number of erroneous attempts involving PQ trees for the solution of au tomatic graph drawing problems that have been presented in the literature have in creased in recent years we present a closer examination of some of the mistakes in order to prevent future research from constructing algorithms with similar errors Throughout this extended abstract we study the computation of maximal...