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### The number of maximal independent sets in a connected graph

Jerrold R. Griggs, Charles M. Grinstead, David R. Guichard
1988 Discrete Mathematics
We determine the maximum on n vertices can have, and we a question of Wilf. number of maximal independent sets which a connected graph completely characterize the extremal graphs, thereby answering Let  ...  Which graphs have this many maximal independent sets? Given any graph G, the subgraph C is a clique if and only the vertices of C form a maximal independent set in the comp!  ...  Wilf [5] found the maximum number of maximal independent sets possible in a tree and asked *what is the maximum number g(n) of maximal independent sets possible in a connected graph on n vertices.  ...

### Split Domination, Independence, and Irredundance in Graphs [article]

Stephen Hedetniemi, Fiona Knoll, Renu Laskar
2016 arXiv   pre-print
A set S of vertices in a graph G =(V,E) is called a split independent set if S is independent and the induced subgraph 〈 V ∖ S 〉 is either disconnected or a K_1.  ...  In 1978, Kulli and Janakiram KulliJanakiramSplit defined the split dominating set: a dominating set S of vertices in a graph G = (V, E) is called split dominating if the induced subgraph 〈 V ∖ S〉 is either  ...  A vertex cut set is a set of vertices S ⊆ V (G) such that G \ S is either a K 1 or a disconnected graph. The connectivity k(G) of a graph G is the size of the smallest vertex cut set of graph G.  ...

### An Alternative Proof of the Largest Number of Maximal Independent Sets in Connected Graphs Having at Most Two Cycles

Min-Jen Jou, Jenq-Jong Lin
2016 Open Journal of Discrete Mathematics
found the maximum number of maximal independent sets in connected graphs which contain at most two cycles.  ...  In this paper, we give an alternative proof to determine the largest number of maximal independent sets among all connected graphs of order n ≥ 12, which contain at most two cycles.  ...  Acknowledgements The authors are grateful to the referees for the helpful comments.  ...

### Combinatorial bounds on connectivity for dominating sets in maximal outerplanar graphs

Santiago Canales, Irene Castro, Gregorio Hernández, Mafalda Martins
2016 Electronic Notes in Discrete Mathematics
In this article we study some variants of the domination concept attending to the connectivity of the subgraph generated by the dominant set. This study is restricted to maximal outerplanar graphs.  ...  We establish tight combinatorial bounds for connected domination, semitotal domination, independent domination and weakly connected domination for any n-vertex maximal outerplaner graph.  ...  The domination number γ(G) is the number of vertices in a smallest dominating set for G.  ...

### Constructing MICDS Algorithm for Data Transmission using Maximal Independent Set in Network Graph

G. N.Purohit, Seema Verma, Usha Sharma, Gargi Chauhan
2014 International Journal of Computer Applications
In this paper, proposed a MICDS (Maximal Independent Connected Dominating Set) algorithm for the WSNs, which are based on the construction of the maximal independent set (MIS) of the representing network  ...  The CDS is a dominating set which induces a connected subgraph. A maximal independent set is an independent set that is not a subset of any other independent set [7, 8] .  ...  Analysis The transmission of the data in the network graph using the Connected Dominating Set (CDS) depends on the maximum degree, independent sets, number of nodes in maximal independent set and hierarchy  ...

### Identifying large sets of unrelated individuals and unrelated markers

Kuruvilla Abraham, Clara Diaz
2014 Source Code for Biology and Medicine
Results: We present FastIndep, a fast random heuristic algorithm for finding a maximal independent set of nodes in an arbitrary undirected graph along with an efficient implementation in C++.  ...  The same methodology can also be used for identifying a suitable set of markers for analyzing population stratification, and other instances where a rapid heuristic for maximal independent sets in large  ...  We thank the Avileña Negra-Iberica Breed Association for providing the genotypes of the cattle population studied in this paper.  ...

### The Number of Maximal Independent Sets in Quasi-Tree Graphs and Quasi-Forest Graphs

Jenq-Jong Lin, Min-Jen Jou
2017 Open Journal of Discrete Mathematics
A connected graph (respectively, graph) G with vertex set ( ) In this paper, we survey on the large numbers of maximal independent sets among all trees, forests, quasi-trees and quasi-forests.  ...  In addition, we further look into the problem of determining the third largest number of maximal independent sets among all quasi-trees and quasi-forests.  ...  The result of the second largest number of maximal independent sets among all connected unicyclic graphs are described in Theorems 2.11.  ...

### The second largest number of maximal independent sets in graphs with at most two cycles

Min-Jen Jou, Jenq-Jong Lin
2016 International Journal of Contemporary Mathematical Sciences
In 2012, Jou settled the second largest number of maximal independent sets in graphs with at most one cycle.  ...  In this paper, we study the second largest number of maximal independent sets among all graphs of order n ≥ 5 with at most two cycles.  ...  There are researches on independent sets in graphs from a different point of view. The Fibonacci number of a graph is the number of independent vertex subsets.  ...

### Connected and Total Vertex covering in Graphs

Sathikala L, Et. al.
2021 Turkish Journal of Computer and Mathematics Education
The minimum cardinality of such a set is called the total vertex covering number and it is denoted by .In this paper a few properties of connected vertex cover and total vertex covers are studied and specific  ...  The minimum cardinality of such a set is called the connected vertex covering number and it is denoted by . A Vertex cover S of G is a total vertex cover if the induced subgraph of S has no isolates.  ...  The number of edges in a smallest maximal independent set of edges in G is denoted by ' 1 () G  . 12.  ...

### Quantum Complexity Bounds for Independent Set Problems [article]

Sebastian Doern
2007 arXiv   pre-print
In particular, we give quantum algorithms for computing a maximal and a maximum independent set in a graph. We present applications of these algorithms for some graph problems.  ...  We present quantum complexity lower and upper bounds for independent set problems in graphs.  ...  A maximal independent set in a graph is an independent set which is contained in no other independent set. A maximum independent set is a largest independent set of a graph G.  ...

### Relatively Prime Graph RPn

U. Kumaran, K. Marimuthu
2018 International Journal of Engineering Science Advanced Computing and Bio-Technology
In this paper we introduce Relatively prime graph of order n , Further we investigate basic nature of this graph and we give an estimation of connectivity number of this graph.  ...  An independent set of G with maximum number of vertices is called a maximum independent set of G . The number of vertices of a maximum independent set of G is called the independence number of G .  ...  set   V G of G is said to be an independent set of G if the induced subgraph of   G U is a trivial graph.  ...

### Edge Cut Domination, Irredundance, and Independence in Graphs [article]

Todd Fenstermacher, Stephen Hedetniemi, Renu Laskar
2016 arXiv   pre-print
An edge dominating set F of a graph G=(V,E) is an edge cut dominating set if the subgraph 〈 V,G-F 〉 is disconnected.  ...  In this paper we study the edge cut domination number and investigate its relationships with other parameters of graphs.  ...  The independent domination number and independence number, i(G) and β (G), are respectively the minimum and maximum cardinalities taken over all maximal sets of independent vertices of G.  ...

### On minimum weakly connected independent sets for wireless sensor networks: properties and enumeration algorithm

Fatiha Bendali, Jean Mailfert, Djelloul Mameri, Jacek Blazewicz, Erwin Pesch, Cynthia Philipps, Denis Trystram, Guochuan Zhang
2014 Reserche operationelle
Given an undirected connected graph G = (V, E), we say that an independent set S in G is weakly connected if the partial graph is the set of edges having exactly one end in S.  ...  In this paper, we study a dominating set that we call Weakly Connected Independent Set (wcis).  ...  In [22] , Moon and Moser showed that the number of maximal independent sets of a graph with n vertices is upper bounded by 1.443 n . Johnson et alii.  ...

### Wireless networking, dominating and packing

Weili Wu, Xiaofeng Gao, Panos M. Pardalos, Ding-Zhu Du
2009 Optimization Letters
Recently, there are many papers published in the literature on study of dominating sets, connected dominating sets and their variations motivated from various applications in wireless networks and social  ...  In this article, we survey those developments for wireless networking, dominating, and packing problems.  ...  Acknowledgment This work is supported in part by National Science Foundation under grant CCF0514796 and CCF0829993.  ...

### Maximal Independent Sets and Maximal Matchings in Series-Parallel and Related Graph Classes

Michael Drmota, Lander Ramos, Clément Requilé, Juanjo Rué, Marc Herbstritt
2018 International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms
We provide combinatorial decompositions as well as asymptotic tight estimates for two maximal parameters: the number and average size of maximal independent sets and maximal matchings in series-parallel  ...  In particular, our results extend previous results of Meir and Moon for trees [Meir, Moon: On maximal independent sets of nodes in trees, Journal of Graph Theory 1988].  ...  Acknowledgements The authors would like to thank the anonymous referees for their careful reading, commentaries and useful suggestions.  ...
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