In numerous practical applications, it is necessary to find the smallest possible tree with a bounded diameter. A diameter-constrained minimum spanning tree (DCMST) of a given undirected, edge-weighted graph, G, is the smallest-weight spanning tree of all spanning trees of G which contain no path with more than k edges, where k is a given positive integer. The problem of finding a DCMST is NP-complete for all values of k; 4 k (n -- 2), except when all edge-weights are identical. A DCMST is essential for the efficiency of various distributed mutual exclusion algorithms, where it can minimize the number of messages communicated among processors per critical section. It is also useful in linear lightwave networks, where it can minimize interference in the network by limiting the traffic in the network lines. Another practical application requiring a DCMST arises in data compression, where some algorithms compress a file utilizing a tree data-structure, and decompress a path in the tree to access a record. A DCMST helps such algorithms to be fast without sacrificing a lot of storage space. We present a survey of the literature on the DCMST problem, study the expected diameter of a random labeled tree, and present five new polynomial-time algorithms for an approximate DCMST. One of our new algorithms constructs an approximate DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to iii the tree in each stage of the construction. Three other new algorithms start with an unconstrained minimum spanning tree, and iteratively refine it into an approximate DCMST. We also present an algorithm designed for the special case when the diameter is required to be no more than 4. Such a diameter-4 tree is also used for evaluating the quality of o...

. An integer-valued graph function ß is an interpolating function if for every connected graph G, ß(T (G)) is a set of consecutive integers, where T (G) is the set of all spanning trees of G. The interpolating character of a number of domination related parameters is considered. 1. Introduction and preliminary results. In 1980, G. Chartrand [4] raised the following problem: If a graph G possesses a spanning tree having m end vertices and another having M end vertices, where M ? m, does G possess a spanning tree having k end vertices for every k between m and M? This question was answered affirmatively in [22] and [1] and it led to a number of papers studying the interpolation properties of parameters of spanning trees of a given graph. In [13], the various known interpolation results are examined and classified on the basis of the proof techniques used in establishing them. Motivated by results of the papers [2], [13] and [15], we investigate interpolation properties of domination rel...