Lower bounds for strictly fundamental cycle bases in grid graphs

Networks ◽  
2009 ◽  
Vol 53 (2) ◽  
pp. 191-205 ◽  
Author(s):  
Ekkehard Köhler ◽  
Christian Liebchen ◽  
Gregor Wünsch ◽  
Romeo Rizzi
Keyword(s):  
Lower Bounds ◽  
Grid Graphs ◽  
Fundamental Cycle ◽  
Cycle Bases

scholarly journals Mathematical models and a constructive heuristic for finding minimum fundamental cycle bases

2005 ◽  
Vol 15 (1) ◽  
pp. 15-24 ◽  
Author(s):  
Leo Liberti ◽  
Edoardo Amaldi ◽  
Francesco Maffioli ◽  
Nelson Maculan
Keyword(s):  
Linear Programming ◽  
Total Cost ◽  
Fundamental Cycle ◽  
Cycle Basis ◽  
Cpu Time ◽  
Constructive Heuristic ◽  
Cycle Bases ◽  
Time Required ◽  
Application Fields

The problem of finding a fundamental cycle basis with minimum total cost in a graph arises in many application fields. In this paper we present some integer linear programming formulations and we compare their performances, in terms of instance size, CPU time required for the solution, and quality of the associated lower bound derived by solving the corresponding continuous relaxations. Since only very small instances can be solved to optimality with these formulations and very large instances occur in a number of applications, we present a new constructive heuristic and compare it with alternative heuristics.

Minimum Weakly Fundamental Cycle Bases Are Hard To Find

Algorithmica ◽  
2007 ◽  
Vol 53 (3) ◽  
pp. 402-424 ◽  
Author(s):  
Romeo Rizzi
Keyword(s):  
Fundamental Cycle ◽  
Cycle Bases

Algorithms for finding minimum fundamental cycle bases in graphs

2004 ◽  
Vol 17 ◽  
pp. 29-33 ◽  
Author(s):  
Edoardo Amaldi ◽  
Leo Liberti ◽  
Francesco Maffioli ◽  
Nelson Maculan
Keyword(s):  
Fundamental Cycle ◽  
Cycle Bases

scholarly journals Total Face Irregularity Strength of Type (Alpha, Beta, Gamma) of Grid Graphs

2021 ◽  
Author(s):  
Abdul Aleem Mughal ◽  
Raja Noshad Jamil
Keyword(s):  
Lower Bounds ◽  
Grid Graphs ◽  
Minimum Integer ◽  
Alpha Beta ◽  
Plane Grid ◽  
Irregularity Strength

We investigate new graph characteristics namely total (vertex, edge) face irregularity strength of gen- eralized plane grid graphs Gmn under k-labeling Phi of type (Alpha, Beta, Gamma). The minimum integer k for which a vertex-edge labelled graph has distinct face weights is called the total face irregularity strength of the graph and is denoted by tfs(Gmn). In this article, the graphs G = (V;E; F) under consideration are simple, finite, undirected and planar. We will estimate the exact tight lower bounds for the total face irregularity strength of some families of generalized plane grid graphs.

scholarly journals Interchangeability of Relevant Cycles in Graphs

10.37236/1494 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Petra M. Gleiss ◽  
Josef Leydold ◽  
Peter F. Stadler
Keyword(s):  
Lower Bounds ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Upper And Lower Bounds ◽  
Cycle Basis ◽  
Cycle Bases

The set ${\cal R}$ of relevant cycles of a graph $G$ is the union of its minimum cycle bases. We introduce a partition of ${\cal R}$ such that each cycle in a class ${\cal W}$ can be expressed as a sum of other cycles in ${\cal W}$ and shorter cycles. It is shown that each minimum cycle basis contains the same number of representatives of a given class ${\cal W}$. This result is used to derive upper and lower bounds on the number of distinct minimum cycle bases. Finally, we give a polynomial-time algorithm to compute this partition.

Efficient Edge-Swapping Heuristics for Finding Minimum Fundamental Cycle Bases

2004 ◽  
pp. 14-29 ◽  
Author(s):  
Edoardo Amaldi ◽  
Leo Liberti ◽  
Nelson Maculan ◽  
Francesco Maffioli
Keyword(s):  
Fundamental Cycle ◽  
Edge Swapping ◽  
Cycle Bases

scholarly journals Total Face Irregularity Strength of Type (Alpha, Beta, Gamma) of Grid Graphs

2021 ◽  
Author(s):  
Abdul Aleem Mughal ◽  
Raja Noshad Jamil
Keyword(s):  
Lower Bounds ◽  
Grid Graphs ◽  
Minimum Integer ◽  
Alpha Beta ◽  
Plane Grid ◽  
Irregularity Strength

We investigate new graph characteristics namely total (vertex, edge) face irregularity strength of gen- eralized plane grid graphs Gmn under k-labeling Phi of type (Alpha, Beta, Gamma). The minimum integer k for which a vertex-edge labelled graph has distinct face weights is called the total face irregularity strength of the graph and is denoted by tfs(Gmn). In this article, the graphs G = (V;E; F) under consideration are simple, finite, undirected and planar. We will estimate the exact tight lower bounds for the total face irregularity strength of some families of generalized plane grid graphs.

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