An optimal pram algorithm for a spanning tree on trapezoid graphs

LetG be a graph withn vertices andm edges. The problem of constructing a spanning tree is to find a connected subgraph ofG withn vertices andn?1 edges. In this paper, we propose anO(logn) time parallel algorithm withO(n/logn), processors on an EREW PRAM for constructing a spanning tree on trapezoid graphs.     

A polynomial algorithm to compute the minimum degree spanning trees of directed acyclic graphs with applications to the broadcast problem

In this paper, we focus on the directed minimum degree spanning tree problem and the minimum time broadcast problem. Firstly, we propose a polynomial time algorithm for the minimum degree spanning tree problem in directed acyclic graphs. The algorithm starts with an arbitrary spanning tree, and iteratively reduces the number of vertices of maximum degree. We can prove that the algorithm must reduce a vertex of the maximum degree for each phase, and finally result in an optimal tree. The algorithm terminates in O(mnlogn) time, where m and n are the numbers of edges and vertices of the graph, respectively. Moreover, we apply the new algorithm to the minimum time broadcast problem. Two consequences for directed acyclic graphs are: (1) the problem under the vertex-disjoint paths mode can be approximated within a factor of of the optimum in O(mnlogn)-time; (2) the problem under the edge-disjoint paths mode can be approximated within a factor of O(Δ^*/logΔ^*) of the optimum in O(mnlogn)-time, where Δ^* is the minimum k such that there is a spanning tree of the graph with maximum degree k.     

Minimum edge ranking spanning trees of split graphs

Given a graph G, the minimum edge ranking spanning tree problem (MERST) is to find a spanning tree of G whose edge ranking is minimum. However, this problem is known to be NP-hard for general graphs. In this paper, we show that the problem MERST has a polynomial time algorithm for split graphs, which have useful applications in practice. The result is also significant in the sense that this is a first non-trivial graph class for which the problem MERST is found to be polynomially solvable. We also show that the problem MERST for threshold graphs can be solved in linear time, where threshold graphs are known to be split.     

Efficient algorithms for a family of matroid intersection problems

Consider a matroid where each element has a real-valued cost and a color, red or green; a base is sought that contains q red elements and has smallest possible cost. An algorithm for the problem on general matroids is presented, along with a number of variations. Its efficiency is demonstrated by implementations on specific matroids. In all cases but one, the running time matches the best-known algorithm for the problem without the red element constraint: On graphic matroids, a smallest spanning tree with q red edges can be found in time O(n log n) more than what is needed to find a minimum spanning tree. A special case is finding a smallest spanning tree with a degree constraint; here the time is only O(m + n) more than that needed to find one minimum spanning tree. On transversal and matching matroids, the time is the same as the best-known algorithms for a minimum cost base. This also holds for transversal matroids for convex graphs, which model a scheduling problem on unit-length jobs with release times and deadlines. On partition matroids, a linear-time algorithm is presented. Finally an algorithm related to our general approach finds a smallest spanning tree on a directed graph, where the given root has a degree constraint. Again the time matches the best-known algorithm for the problem without the red element (i.e., degree) constraint.     

A Generalization of Heterochromatic Graphs and f-Chromatic Spanning Forests

In 2006, Suzuki, and Akbari and Alipour independently presented a necessary and sufficient condition for edge-colored graphs to have a heterochromatic spanning tree, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors. In this paper, we propose f-chromatic graphs as a generalization of heterochromatic graphs. An edge-colored graph is f-chromatic if each color c appears on at most f(c) edges. We also present a necessary and sufficient condition for edge-colored graphs to have an f-chromatic spanning forest with exactly m components. Moreover, using this criterion, we show that a g-chromatic graph G of order n with ${|E(G)| > \binom{n-m}{2}}$ has an f-chromatic spanning forest with exactly m (1 ≤ m ≤ n ? 1) components if ${g(c) \le \frac{|E(G)|}{n-m}f(c)}$ for any color c.     

On the Minimum Average Distance Spanning Tree of the Hypercube

Given an undirected and connected graph G, with a non-negative weight on each edge, the Minimum Average Distance (MAD) spanning tree problem is to find a spanning tree of G which minimizes the average distance between pairs of vertices. This network design problem is known to be NP-hard even when the edge-weights are equal. In this paper we make a step towards the proof of a conjecture stated by A.A. Dobrynin, R. Entringer and I. Gutman in 2001, and which says that the binomial tree B _n is a MAD spanning tree of the hypercube H _n . More precisely, we show that the binomial tree B _n is a local optimum with respect to the 1-move heuristic which, starting from a spanning tree T of the hypercube H _n , attempts to improve the average distance between pairs of vertices, by adding an edge e of H _n -T and removing an edge e′ from the unique cycle created by e. We also present a greedy algorithm which produces good solutions for the MAD spanning tree problem on regular graphs such as the hypercube and the torus.     

Spanning Star Trees in Regular Graphs

For a subset W of vertices of an undirected graph G, let S(W) be the subgraph consisting of W, all edges incident to at least one vertex in W, and all vertices adjacent to at least one vertex in W. If S(W) is a tree containing all the vertices of G, then we call it a spanning star tree of G. In this case W forms a weakly connected but strongly acyclic dominating set for G. We prove that for every r ≥ 3, there exist r-regular n-vertex graphs that have spanning star trees, and there exist r-regular n-vertex graphs that do not have spanning star trees, for all n sufficiently large (in terms of r). Furthermore, the problem of determining whether a given regular graph has a spanning star tree is NP-complete.     

Broadcasting secure messages via optimal independent spanning trees in folded hypercubes

Fault-tolerant broadcasting and secure message distribution are important issues for network applications. It is a common idea to design multiple spanning trees with a specific property in the underlying graph of a network to serve as a broadcasting scheme or a distribution protocol for receiving high levels of fault-tolerance and security. An n-dimensional folded hypercube, denoted by FQ_n, is a strengthening variation of hypercube by adding additional links between nodes that have the furthest Hamming distance. In, [12], Ho(1990) proposed an algorithm for constructing n+1 edge-disjoint spanning trees each with a height twice the diameter of FQ_n. Yang et al. (2009), [29] recently proved that Ho’s spanning trees are indeed independent, i.e., any two spanning trees have the same root, say r, and for any other node v≠r, the two different paths from v to r, one path in each tree, are internally node-disjoint. In this paper, we provide another construction scheme to produce n+1 independent spanning trees of FQ_n, where the height of each tree is equal to the diameter of FQ_n plus one. As a result, the heights of independent spanning trees constructed in this paper are shown to be optimal.     

On encodings of spanning trees

Deo and Micikevicius recently gave a new bijection for spanning trees of complete bipartite graphs. In this paper we devise a generalization of Deo and Micikevicius's method, which is also a modification of Olah's method for encoding the spanning trees of any complete multipartite graph K(n₁,…,n_r). We also give a bijection between the spanning trees of a planar graph and those of any of its planar duals. Finally we discuss the possibility of bijections for spanning trees of DeBriujn graphs, cubes, and regular graphs such as the Petersen graph that have integer eigenvalues.     

An O(m log n) algorithm for the max + sum spanning tree problem

In a graph in which each edge has two weights, the max + sum spanning tree problem seeks a spanning tree that has the minimum value for the combined total of the maximum of one of the edge weights and the sum of the other weights among all the spanning trees in the graph. Exploiting an efficient data structure, an O(m log n) algorithm is presented for solving this problem. This improves the currently known bound of O(mn).     

Generating 3-vertex connected spanning subgraphs

In this paper we present an algorithm to generate all minimal 3-vertex connected spanning subgraphs of an undirected graph with n vertices and m edges in incremental polynomial time, i.e., for every K we can generate K (or all) minimal 3-vertex connected spanning subgraphs of a given graph in O(K²log(K)m²+K²m³) time, where n and m are the number of vertices and edges of the input graph, respectively. This is an improvement over what was previously available and is the same as the best known running time for generating 2-vertex connected spanning subgraphs. Our result is obtained by applying the decomposition theory of 2-vertex connected graphs to the graphs obtained from minimal 3-vertex connected graphs by removing a single edge.     

Locally connected spanning trees in strongly chordal graphs and proper circular-arc graphs

A locally connected spanning tree of a graph G is a spanning tree T of G such that the set of all neighbors of v in T induces a connected subgraph of G for every v∈V(G). The purpose of this paper is to give linear-time algorithms for finding locally connected spanning trees on strongly chordal graphs and proper circular-arc graphs, respectively.     

Spanning 3-ended trees in k-connected K 1,4-free graphs

A tree with at most m leaves is called an m-ended tree.Kyaw proved that every connected K1,4-free graph withσ4(G)n-1 contains a spanning 3-ended tree.In this paper we obtain a result for k-connected K1,4-free graphs with k 2.Let G be a k-connected K1,4-free graph of order n with k 2.Ifσk+3(G)n+2k-2,then G contains a spanning 3-ended tree.     

Tree 3-spanners in 2-sep directed path graphs: Characterization, recognition, and construction

A spanning tree T of a graph G is called a treet-spanner, if the distance between any two vertices in T is at most t-times their distance in G. A graph that has a tree t-spanner is called a treet-spanner admissible graph. The problem of deciding whether a graph is tree t-spanner admissible is NP-complete for any fixed t≥4, and is linearly solvable for t=1 and t=2. The case t=3 still remains open. A directed path graph is called a 2-sep directed path graph if all of its minimal a−b vertex separator for every pair of non-adjacent vertices a and b are of size two. Le and Le [H.-O. Le, V.B. Le, Optimal tree 3-spanners in directed path graphs, Networks 34 (2) (1999) 81-87] showed that directed path graphs admit tree 3-spanners. However, this result has been shown to be incorrect by Panda and Das [B.S. Panda, Anita Das, On tree 3-spanners in directed path graphs, Networks 50 (3) (2007) 203-210]. In fact, this paper observes that even the class of 2-sep directed path graphs, which is a proper subclass of directed path graphs, need not admit tree 3-spanners in general. It, then, presents a structural characterization of tree 3-spanner admissible 2-sep directed path graphs. Based on this characterization, a linear time recognition algorithm for tree 3-spanner admissible 2-sep directed path graphs is presented. Finally, a linear time algorithm to construct a tree 3-spanner of a tree 3-spanner admissible 2-sep directed path graph is proposed.     

A GRASP algorithm for the multi-criteria minimum spanning tree problem

This paper proposes a GRASP (Greedy Randomized Adaptive Search Procedure) algorithm for the multi-criteria minimum spanning tree problem, which is NP-hard. In this problem a vector of costs is defined for each edge of the graph and the problem is to find all Pareto optimal or efficient spanning trees (solutions). The algorithm is based on the optimization of different weighted utility functions. In each iteration, a weight vector is defined and a solution is built using a greedy randomized constructive procedure. The found solution is submitted to a local search trying to improve the value of the weighted utility function. We use a drop-and-add neighborhood where the spanning trees are represented by Prufer numbers. In order to find a variety of efficient solutions, we use different weight vectors, which are distributed uniformly on the Pareto frontier. The proposed algorithm is tested on problems with r=2 and 3 criteria. For non-complete graphs with n=10, 20 and 30 nodes, the performance of the algorithm is tested against a complete enumeration. For complete graphs with n=20, 30 and 50 nodes the performance of the algorithm is tested using two types of weighted utility functions. The algorithm is also compared with the multi-criteria version of the Kruskal’s algorithm, which generates supported efficient solutions. This work was funded by the Municipal Town Hall of Campos dos Goytacazes city. The used computer was acquired with resource of CNPq.     

Tree 3-spanners in 2-sep chordal graphs: Characterization and algorithms

A spanning tree T of a graph G is said to be a treet-spanner if the distance between any two vertices in T is at most t times their distance in G. A graph that has a tree t-spanner is called a treet-spanner admissible graph. The problem of deciding whether a graph is tree t-spanner admissible is NP-complete for any fixed t≥4 and is linearly solvable for t≤2. The case t=3 still remains open. A chordal graph is called a 2-sep chordal graph if all of its minimal a−b vertex separators for every pair of non-adjacent vertices a and b are of size two. It is known that not all 2-sep chordal graphs admit tree 3-spanners. This paper presents a structural characterization and a linear time recognition algorithm of tree 3-spanner admissible 2-sep chordal graphs. Finally, a linear time algorithm to construct a tree 3-spanner of a tree 3-spanner admissible 2-sep chordal graph is proposed.     

Tree spanners in planar graphs

A tree t-spanner of a graph G is a spanning subtree T of G in which the distance between every pair of vertices is at most t times their distance in G. Spanner problems have received some attention, mostly in the context of communication networks. It is known that for general unweighted graphs, the problem of deciding the existence of a tree t-spanner can be solved in polynomial time for t=2, while it is NP-hard for any t⩾4; the case t=3 is open, but has been conjectured to be hard. In this paper, we consider tree spanners in planar graphs. We show that even for planar unweighted graphs, it is NP-hard to determine the minimum t for which a tree t-spanner exists. On the other hand, we give a polynomial algorithm for any fixed t that decides for planar unweighted graphs with bounded face length whether there is a tree t-spanner. Furthermore, we prove that it can be decided in polynomial time whether a planar unweighted graph has a tree t-spanner for t=3.     

An optimal algorithm for finding depth-first spanning tree on permutation graphs

LetG be a connected graph ofn vertices. The problem of finding a depth-first spanning tree ofG is to find a connected subgraph ofG with then vertices andn − 1 edges by depth-first-search. In this paper, we propose anO(n) time algorithm to solve this problem on permutation graphs.     

Star-Uniform Graphs

A star-factor of a graph is a spanning subgraph each of whose components is a star. A graph G is called star-uniform if all star-factors of G have the same number of components. Motivated by the minimum cost spanning tree and the optimal assignment problems, Hartnell and Rall posed an open problem to characterize all the star-uniform graphs. In this paper, we show that a graph G is star-uniform if and only if G has equal domination and matching number. From this point of view, the star-uniform graphs were characterized by Randerath and Volkmann. Unfortunately, their characterization is incomplete. By deploying Gallai–Edmonds Matching Structure Theorem, we give a clear and complete characterization of star-unform graphs.     

An implicit representation of chordal comparability graphs in linear time

Ma and Spinrad have shown that every transitive orientation of a chordal comparability graph is the intersection of four linear orders. That is, chordal comparability graphs are comparability graphs of posets of dimension four. Among other uses, this gives an implicit representation of a chordal comparability graph using O(n) integers so that, given two vertices, it can be determined in O(1) time whether they are adjacent, no matter how dense the graph is. We give a linear time algorithm for finding the four linear orders, improving on their bound of O(n²).