scholarly journals Eigenvalues, traceability, and perfect matching of a graph

2021 ◽  
Vol 57 (3) ◽  
pp. 274-285
Author(s):  
V. I. Benediktovich
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Perfect Matching ◽  
Hamiltonian Path ◽  
Recognition Algorithm ◽  
Recognition Problem ◽  
Cyclomatic Number ◽  
Np Complete

It is well known that the recognition problem of the existence of a perfect matching in a graph, as well as the recognition problem of its Hamiltonicity and traceability, is NP-complete. Quite recently, lower bounds for the size and the spectral radius of a graph that guarantee the existence of a perfect matching in it have been obtained. We improve these bounds, firstly, by using the available bounds for the size of the graph for existence of a Hamiltonian path in it, and secondly, by finding new lower bounds for the spectral radius of the graph that are sufficient for the traceability property. Moreover, we develop the recognition algorithm of the existence of a perfect matching in a graph. This algorithm uses the concept of a (κ,τ)-regular set, which becomes polynomial in the class of graphs with a fixed cyclomatic number.

scholarly journals Gray codes avoiding matchings

2009 ◽  
Vol Vol. 11 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Darko Dimitrov ◽  
Tomáš Dvořák ◽  
Petr Gregor ◽  
Riste Škrekovski
Keyword(s):  
Hamiltonian Cycle ◽  
Perfect Matching ◽  
Hamiltonian Path ◽  
Gray Code ◽  
Gray Codes ◽  
Similar Characterization ◽  
International Audience ◽  
Forbidden Configurations ◽  
Np Complete ◽  
Necessary And Sufficient

Graphs and Algorithms International audience A (cyclic) n-bit Gray code is a (cyclic) ordering of all 2(n) binary strings of length n such that consecutive strings differ in a single bit. Equivalently, an n-bit Gray code can be viewed as a Hamiltonian path of the n-dimensional hypercube Q(n), and a cyclic Gray code as a Hamiltonian cycle of Q(n). In this paper we study (cyclic) Gray codes avoiding a given set of faulty edges that form a matching. Given a matching M and two vertices u, v of Q(n), n >= 4, our main result provides a necessary and sufficient condition, expressed in terms of forbidden configurations for M, for the existence of a Gray code between u and v that avoids M. As a corollary. we obtain a similar characterization for a cyclic Gray code avoiding M. In particular, in the case that M is a perfect matching, Q(n) has a (cyclic) Gray code that avoids M if and only if Q(n) - M is a connected graph. This complements a recent result of Fink, who proved that every perfect matching of Q(n) can be extended to a Hamiltonian cycle. Furthermore, our results imply that the problem of Hamilionicity of Q(n) with faulty edges, which is NP-complete in general, becomes polynomial for up to 2(n-1) edges provided they form a matching.

DIMMA-Implemented Metaheuristics for Finding Shortest Hamiltonian Path Between Iranian Cities Using Sequential DOE Approach for Parameters Tuning

2012 ◽  
pp. 289-305
Author(s):  
Masoud Yaghini ◽  
Mohsen Momeni ◽  
Mohammadreza Sarmadi
Keyword(s):  
Hamiltonian Path ◽  
Search Algorithms ◽  
Sequential Design ◽  
Cpu Time ◽  
Local Search Methods ◽  
Parameters Tuning ◽  
Efficiency And Effectiveness ◽  
Np Complete ◽  
First Time

A Hamiltonian path is a path in an undirected graph, which visits each node exactly once and returns to the starting node. Finding such paths in graphs is the Hamiltonian path problem, which is NP-complete. In this paper, for the first time, a comparative study on metaheuristic algorithms for finding the shortest Hamiltonian path for 1071 Iranian cities is conducted. These are the main cities of Iran based on social-economic characteristics. For solving this problem, four hybrid efficient and effective metaheuristics, consisting of simulated annealing, ant colony optimization, genetic algorithm, and tabu search algorithms, are combined with the local search methods. The algorithms’ parameters are tuned by sequential design of experiments (DOE) approach, and the most appropriate values for the parameters are adjusted. To evaluate the proposed algorithms, the standard problems with different sizes are used. The performance of the proposed algorithms is analyzed by the quality of solution and CPU time measures. The results are compared based on efficiency and effectiveness of the algorithms.

DIMMA-Implemented Metaheuristics for Finding Shortest Hamiltonian Path Between Iranian Cities Using Sequential DOE Approach for Parameters Tuning

2011 ◽  
Vol 2 (2) ◽  
pp. 74-92 ◽  
Author(s):  
Masoud Yaghini ◽  
Mohsen Momeni ◽  
Mohammadreza Sarmadi
Keyword(s):  
Social Economic ◽  
Undirected Graph ◽  
Hamiltonian Path ◽  
Sequential Design ◽  
Local Search Methods ◽  
Parameters Tuning ◽  
Efficiency And Effectiveness ◽  
Np Complete ◽  
First Time

A Hamiltonian path is a path in an undirected graph, which visits each node exactly once and returns to the starting node. Finding such paths in graphs is the Hamiltonian path problem, which is NP-complete. In this paper, for the first time, a comparative study on metaheuristic algorithms for finding the shortest Hamiltonian path for 1071 Iranian cities is conducted. These are the main cities of Iran based on social-economic characteristics. For solving this problem, four hybrid efficient and effective metaheuristics, consisting of simulated annealing, ant colony optimization, genetic algorithm, and tabu search algorithms, are combined with the local search methods. The algorithms’ parameters are tuned by sequential design of experiments (DOE) approach, and the most appropriate values for the parameters are adjusted. To evaluate the proposed algorithms, the standard problems with different sizes are used. The performance of the proposed algorithms is analyzed by the quality of solution and CPU time measures. The results are compared based on efficiency and effectiveness of the algorithms.

scholarly journals On the König deficiency of zero-reducible graphs

2019 ◽  
Vol 39 (1) ◽  
pp. 273-292
Author(s):  
Miklós Bartha ◽  
Miklós Krész
Keyword(s):  
Perfect Matching ◽  
Linear Time ◽  
Perfect Matchings ◽  
Covering Number ◽  
Matching Number ◽  
Empty Graph ◽  
Reduction System ◽  
Vertex Covering ◽  
Np Complete ◽  
The Difference

Abstract A confluent and terminating reduction system is introduced for graphs, which preserves the number of their perfect matchings. A union-find algorithm is presented to carry out reduction in almost linear time. The König property is investigated in the context of reduction by introducing the König deficiency of a graph G as the difference between the vertex covering number and the matching number of G. It is shown that the problem of finding the König deficiency of a graph is NP-complete even if we know that the graph reduces to the empty graph. Finally, the König deficiency of graphs G having a vertex v such that $$G-v$$G-v has a unique perfect matching is studied in connection with reduction.

scholarly journals Some lower bounds for the spectral radius of matrices using traces

2010 ◽  
Vol 432 (4) ◽  
pp. 1007-1016 ◽  
Author(s):  
Lin Wang ◽  
Mao-Zhi Xu ◽  
Ting-Zhu Huang
Keyword(s):  
Spectral Radius ◽  
Lower Bounds

Tight Lower Bounds on the Resolution Complexity of Perfect Matching Principles

2016 ◽  
Vol 145 (3) ◽  
pp. 229-242 ◽  
Author(s):  
Dmitry Itsykson ◽  
Vsevolod Oparin ◽  
Mikhail Slabodkin ◽  
Dmitry Sokolov
Keyword(s):  
Lower Bounds ◽  
Perfect Matching

scholarly journals Cauchy's interlace theorem and lower bounds for the spectral radius

2000 ◽  
Vol 23 (8) ◽  
pp. 563-566 ◽  
Author(s):  
A. McD. Mercer ◽  
Peter R. Mercer
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Lower Bounds ◽  
Simple Proof ◽  
Symmetric Matrix ◽  
Real Symmetric Matrix

We present a short and simple proof of the well-known Cauchy interlace theorem. We use the theorem to improve some lower bound estimates for the spectral radius of a real symmetric matrix.

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