Absrracr -In l!X3, Char [4] presented an algorithm to enumerate all the spanning trees of an undirected graph G. This algorithm starts with a known initial spanning tree of G, and generates all the other spanning trees along with certain spanning non-tree subgraphs of G. In this paper a detailed complexity analysis of char’s algorithm and methods to speed up the algorithm are discussed. Two heu...

The linear or cyclic cutwidth of a graph G is the minimum congestion when G is embedded into either a path or a cycle respectively. A graph is cutwith critical if it is homeomorphically minimal and all of its subgraphs have lower cutwitdth. Our purpose is to extend the study of congestion critical graphs to embeddings on spanning trees.

We study a problem in a network. The input is an edgeweighted graph G = (V, E) such that V contains a specific source node r. Every v ∈ V \{r} is an entity which wants to be connected to r either directly or via other entities. The main question is how do the entities deviate from a socially optimal network if they are not monitored by a central authority. We provide theoretical bounds on the (...

In 2001, Brualdi and Hollingsworth proved that an edge-colored balanced complete bipartite graph Kn,n with a color set C = {1, 2, 3, . . . , 2n − 1} has a heterochromatic spanning tree if the number of edges colored with colors in R is more than |R|/4 for any non-empty subset R ⊆ C, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors, namely, any color appe...

This paper studies a 4-approximation algorithm for k-prize collecting Steiner tree problems. This problem generalizes both k-minimum spanning tree problems and prize collecting Steiner tree problems. Our proposed algorithm employs two 2-approximation algorithms for k-minimum spanning tree problems and prize collecting Steiner tree problems. Also our algorithm framework can be applied to a speci...

We demonstrate a collection of techniques that seek to make the transition between drawings based on two topologically distinct spanning trees of the same graph as clear as possible. As Herman, Melançon, and Marshall note [HMM00], one way to draw a large graph is to extract a spanning tree from it, use a tree layout algorithm [CK95, Ead92, RT81, II90, TM02, GADM04, LY05] to draw the spanning tr...