The building cost of a spanning tree is the sum of weights of the edges used to construct the spanning tree. The routing cost of a source vertex s on a spanning tree T is the total summation of distances between the source vertex s and all the vertices d in T . Given a source vertices set S, the multiple-source routing cost of a spanning tree T is the summation of the routing costs for source v...

A geometric graph is a graph whose vertices are points in the plane and whose edges are straight-line segments between the points. A plane spanning tree in a geometric graph is a spanning tree that is non-crossing. Let R and B be two disjoint sets of points in the plane where the points of R are colored red and the points of B are colored blue, and let n = |R ∪ B|. A bichromatic plane spanning ...

Halin conjectured 20 years ago that a graph has normal spanning tree if and only every minor of it countable colouring number. We prove Halin’s conjecture. This implies forbidden characterisation for the property having tree.

Given a graph G with a speciied root node r. A spanning tree in G where each node has distance at most 2 from r is called a 2-hop spanning tree. For given edge weights the 2-hop spanning tree problem is to nd a minimum weight 2-hop spanning tree. The problem is NP-hard and has some interesting applications. We study a polytope associated with a directed model of the problem give a completeness ...

The Krushkal polynomial is the most general polynomial invariant of graphs on surfaces. It was introduced in [1]. A collection of edges forms a spanning quasi-tree if the regular neighborhood of the corresponding spanning subgraph has one boundary component. In the case of plane graphs it is just a spanning tree. The set of edges of a spanning quasi-trees is a feasible set of the corresponding ...