To determine if a graph has a spanning tree with few leaves is NP-hard. In this paper we study the parametric dual of this problem, k-INTERNAL SPANNING TREE (Does G have a spanning tree with at least k internal vertices?). We give an algorithm running in time O(2 log k · k + k · n). We also give a 2-approximation algorithm for the problem. However, the main contribution of this paper is that we...

We propose an univesal scheme to design loop-free and superstabilizing protocols for constructing spanning trees optimizing any tree metrics (not only those that are isomorphic to a shortest path tree). Our scheme combines a novel super-stabilizing loop-free BFS with an existing self-stabilizing spanning tree that optimizes a given metric. The composition result preserves the best properties of...

A spanning tree in a connected graph G is a subgraph that contains every vertex of G and is itself a tree. Clearly, if G is a tree then it has only one spanning tree. Every connected graph contains at least one spanning tree: iteratively remove an edge from any cycle that is present until the graph contains no cycles. Counting spanning trees is a very natural problem. Following Lyons [5] we wil...

The max tree-height of an undirected graph is the longest possible length of a path among all spanning trees of the graph. A maximum height spanning tree of an undirected graph is a spanning tree that has a path of length equal to the max tree-height of the graph. Finding the max tree-height of a graph, or similarly some spanning tree of maximum height, is an NP-hard optimization problem for wh...