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### Algorithms for Maximum Internal Spanning Tree Problem for Some Graph Classes [article]

Gopika Sharma, Arti Pandey, Michael C. Wigal
2021 arXiv   pre-print
For a given graph G, a maximum internal spanning tree of G is a spanning tree of G with maximum number of internal vertices.  ...  The Maximum Internal Spanning Tree (MIST) problem is to find a maximum internal spanning tree of the given graph. The MIST problem is a generalization of the Hamiltonian path problem.  ...  Algorithms for Maximum Internal Spanning Tree Problem for Some Graph Classes *  ...

### Grafalgo - A Library of Graph Algorithms and Supporting Data Structures (revised) [article]

Jonathan Turner
2016 arXiv   pre-print
This report provides an (updated) overview of Grafalgo, an open-source library of graph algorithms and the data structures used to implement them.  ...  The programs in this library were originally written to support a graduate class in advanced data structures and algorithms at Washington University.  ...  Minimum Spanning Tree Problem The objective of the minimum spanning tree problem is to find a spanning tree of an edge-weighted graph that has the smallest total weight.  ...

### Approximating node-weighted multicast trees in wireless ad-hoc networks

Thomas Erlebach, Ambreen Shahnaz
2009 Proceedings of the 2009 International Conference on Wireless Communications and Mobile Computing Connecting the World Wirelessly - IWCMC '09
For graph classes that admit spanning trees of maximum degree at most d, we obtain a 0.775d-approximation algorithm.  ...  We show that this result implies a 3.875-approximation algorithm for unit disk graphs, an O(1/α 2 )-approximation algorithm for α-unit disk graphs, and an O(λ)-approximation algorithm for (λ + 1)claw-free  ...  In this paper, we present a 0.775d-approximation algorithm for the problem of computing a Steiner tree with minimum total weight of the Steiner nodes for graph classes that admit spanning trees of maximum  ...

### Page 67 of Mathematical Reviews Vol. , Issue 2003A [page]

2003 Mathematical Reviews
A recent algorithm for the maximum clique problem is here used as a basis for developing an algorithm for the weighted case.  ...  derive upper and lower bounds for the expected cost of the optimal spanning tree.  ...

### Minimum Connected Dominating Set for Certain Circulant Networks

N. Parthiban, Indra Rajasingh, R. Sundara Rajan
2015 Procedia Computer Science
In general, the problem is proved to be NP-hard. In this paper we find a Minimum Connected Dominating Set for certain Circulant Networks.  ...  The Maximum Leaf Spanning Tree (MLST) problem is to find a spanning tree of a graph with as many leaves as possible.  ...  Given a graph G, the Maximum Leaf Spanning Tree problem is to find the maximum number of vertices of degree one in a spanning tree over all spanning trees of G. Corresponding author.  ...

### Page 3697 of Mathematical Reviews Vol. , Issue 2002E [page]

2002 Mathematical Reviews
For example, if the property is that the edges form a spanning tree, then the dispersion problem becomes the max-min problem: Choose p vertices to maximize the weight of the minimum spanning tree on the  ...  Some problems are apparently considered here for the first time.  ...

### Spanning tree with many leaves in quadratic graph

Akanksa Rastogi, Vinay Singhal
2013 International Journal of Research Studies in Computing
A polynomial algorithm for constructing full spanning trees for 4-regular (Quadratic) graph is presented.  ...  For a connected graph G let L(G) denote the maximum number of leaves in any spanning tree of G.  ...  A constructive proof that all graphs in a particular class have spanning trees with at least m leaves becomes an algorithm to produce such a tree for graphs in this class.  ...

### Either/Or: Using Vertex Cover Structure in Designing FPT-Algorithms — the Case of k-Internal Spanning Tree [chapter]

Elena Prieto, Christian Sloper
2003 Lecture Notes in Computer Science
We show how to use this inherent vertex cover structure to design algorithms for FPT problems, here illustrated mainly by k-INTERNAL SPANNING TREE.  ...  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?).  ...  Acknowledgements: We would like to thank Mike Fellows, Andrzej Proskurowski and Frances Rosamond for very helpful conversations and encouragement.  ...

### Approximation Algorithms for Degree-Constrained Minimum-Cost Network-Design Problems [chapter]

R. Ravi, Madhav V. Marathe, S. S. Ravi, Daniel J. Rosenkrantz, Harry B. Hunt
2009 Fundamental Problems in Computing
A prototypical example is the degree-constrained node-weighted Steiner tree problem: We are given an undirected graph with a non-negative integral function that specifies an upper bound ¥ on the degree  ...  We study network-design problems with two different design objectives: the total cost of the edges and nodes in the network and the maximum degree of any node in the network.  ...  Acknowledgments: We thank the referee for several valuable suggestions. We gratefully acknowledge helpful conversations with M. X. Goemans, P. N. Klein, G. Konjevod, S. Krumke, B. Raghavachari, V. S.  ...

### Blow-Ups, Win/Win's, and Crown Rules: Some New Directions in FPT [chapter]

Michael R. Fellows
2003 Lecture Notes in Computer Science
This survey reviews the basic notions of parameterized complexity, and describes some new approaches to designing FPT algorithms and problem reductions for graph problems.  ...  It applies to the following problem: Max Internal Spanning Tree Input: A graph G and an integer k; Parameter: k; Question: Does G have a spanning tree with at least k internal vertices?  ...  A win/win approach to designing an FPT algorithm for Max Internal Spanning Tree has recently been described by Prieto and Sloper [PS03] .  ...

### Better approximation algorithms for maximum weight internal spanning trees in cubic graphs and claw-free graphs [article]

2020 arXiv   pre-print
Given a connected vertex-weighted graph G, the maximum weight internal spanning tree (MaxwIST) problem asks for a spanning tree of G that maximizes the total weight of internal nodes.  ...  For claw-free graphs of degree at least three, we present an algorithm that computes a spanning tree whose total internal weight is at least 3/5-1/n times the total vertex weight.  ...  There exists a linear-time 3 4 − -approximation algorithm for the maximum weight internal spanning tree problem in cubic graphs, for any > 0.  ...

### Model-checking for successor-invariant first-order formulas on graph classes of bounded expansion

Jan van den Heuvel, Stephan Kreutzer, Michal Pilipczuk, Daniel A. Quiroz, Roman Rabinovich, Sebastian Siebertz
2017 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
In this work we show that the model-checking problem for successor-invariant formulas is fixed-parameter tractable on any class of finite structures whose underlying Gaifman graphs form a class of bounded  ...  It was shown earlier that adding successor-invariance to first-order logic essentially comes at no extra cost for the model-checking problem on classes of finite structures whose underlying Gaifman graph  ...  [11] have shown that adding a spanning tree of constant maximum degree is enough to be able to interpret some successor relation.  ...

### Model-Checking for Successor-Invariant First-Order Formulas on Graph Classes of Bounded Expansion [article]

Jan van den Heuvel and Stephan Kreutzer and Michał Pilipczuk and Daniel A. Quiroz and Roman Rabinovich and Sebastian Siebertz
2017 arXiv   pre-print
In this work we show that the model-checking problem for successor-invariant formulas is fixed-parameter tractable on any class of finite structures whose underlying Gaifman graphs form a class of bounded  ...  It was shown earlier that adding successor-invariance to first-order logic essentially comes at no extra cost for the model-checking problem on classes of finite structures whose underlying Gaifman graph  ...  [11] have shown that adding a spanning tree of constant maximum degree is enough to be able to interpret some successor relation.  ...

### Page 7572 of Mathematical Reviews Vol. , Issue 98M [page]

1998 Mathematical Reviews
In this paper, we propose an algorithm for listing all directed spanning trees of G.  ...  Summary: “A directed spanning tree in a directed graph G = (V,A) is a spanning tree such that no two arcs share their tails.  ...

### Kernelization for Maximum Leaf Spanning Tree with Positive Vertex Weights

Bart M. P. Jansen
2012 Journal of Graph Algorithms and Applications
a spanning tree T for G exists such that the combined weight of the leaves of T is at least k.  ...  In this paper we consider a natural generalization of the wellknown Max Leaf Spanning Tree problem.  ...  for inspiring me to study spanning tree problems.  ...
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