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Computing A DiameterConstrained Minimum Spanning Tree
, 2001
"... In numerous practical applications, it is necessary to find the smallest possible tree with a bounded diameter. A diameterconstrained minimum spanning tree (DCMST) of a given undirected, edgeweighted graph, G, is the smallestweight spanning tree of all spanning trees of G which contain no path wi ..."
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In numerous practical applications, it is necessary to find the smallest possible tree with a bounded diameter. A diameterconstrained minimum spanning tree (DCMST) of a given undirected, edgeweighted graph, G, is the smallestweight 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 NPcomplete for all values of k; 4 k (n  2), except when all edgeweights 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 datastructure, 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 polynomialtime 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 diameter4 tree is also used for evaluating the quality of o...
Interpolation Theorems for Domination Numbers of a Graph
"... . An integervalued 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 ..."
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. An integervalued 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...
Interpolation theorems for the (r, s)domination number of spanning trees
"... If G is a graph without isolated vertices, and if rand s are positive integers, then the (r, s)domination number 'Yr,s(G) of G is the cardinality of a smallest vertex set D such that every vertex not in D is within distance r from some vertex in D, while every vertex in D is within distance s from ..."
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If G is a graph without isolated vertices, and if rand s are positive integers, then the (r, s)domination number 'Yr,s(G) of G is the cardinality of a smallest vertex set D such that every vertex not in D is within distance r from some vertex in D, while every vertex in D is within distance s from another vertex in D. This generalizes the total domination number 'Yt(G) = 'Yl,l(G). Let T ( G) denote the set of all spanning trees of a connected graph G. We prove that 'Yr,s(T(G)) is a set of consecutive integers for every connected graph G of order at least two when s 2': 2r + 1. This is not true if 1:::; s:::; 2r1, and for s = 2r the problem is open. We prove that 'Yr,2r(T ( G)) is a set of consecutive integers for r = 1 and we conjecture this also holds for r 2': 2. We also prove that 'Yr,s(T(G)) is a set of consecutive integers for every 2connected graph G and for any two positive integers rand s.
SWITCHINGS, REALIZATIONS, AND INTERPOLATION THEOREMS FOR GRAPH PARAMETERS
, 2005
"... Interpolation theorems on several graph parameters obtained in the past few years will be reviewed in this paper. Some simplified proofs are provided. Open problems in this direction are reviewed. 1. ..."
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Interpolation theorems on several graph parameters obtained in the past few years will be reviewed in this paper. Some simplified proofs are provided. Open problems in this direction are reviewed. 1.
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 46 (2010), Pages 3–12 Interpolation properties of domination parameters of a graph
"... An integervalued graph function π is an interpolating function if a set π(T (G)) = {π(T):T ∈T(G)} consists of consecutive integers, where T (G) is the set of all spanning trees of a connected graph G. Weconsider the interpolation properties of domination related parameters. 1 ..."
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An integervalued graph function π is an interpolating function if a set π(T (G)) = {π(T):T ∈T(G)} consists of consecutive integers, where T (G) is the set of all spanning trees of a connected graph G. Weconsider the interpolation properties of domination related parameters. 1