An Annotated Glossary of Graph Theory Parameters, with Conjectures

Gera, Ralucca; Haynes, Teresa W.; Hedetniemi, Stephen T.; Henning, Michael A.

doi:10.1007/978-3-319-97686-0_14

Cited by 9 publications

(4 citation statements)

References 395 publications

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“…The terminology of graph theory is consistent with [Harary, 1969;Diestel, 1997;Gera et al, 2016;Karpov, 2017;Gera et al, 2018]. We shall give a formal model for one of the possible simplified variants, when:…”

Section: The Formal Graph Modelmentioning

confidence: 99%

Greedy and branches-and-boundaries methods for the optimal choice of a subset of vertices in a large communication network

Melnikov

Terentyeva²

2023

CAP

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The problems of the proposed paper are generated by the actual tasks of communication networks. The development of communication resources is accompanied by an increase in the dimension of existing communication networks, for which the usual tools for solving network problems are becoming increasingly ineffective.At the same time, the spectrum of communication network problems covers both classical graph theory problems and specialized problems linking them with mathematical models of various fields of mathematics, including optimization theory, dynamic programming, probability theory and the theory of heuristic algorithms. This paper is devoted to the study of the problem of optimal allocation of a certain resource on the objects of the communication network. In this case, a complete set of specialized equipment at the nodes of the communication network is considered as a resource. It is necessary to optimize the number of pieces of equipment with the condition that all fiber-optic communication lines of the network are under the control of installed reflectometers. To solve this problem, variants of two methods were described: the greedy algorithm and the method of branches and boundaries. On the basis of the described algorithms, the computer programs were implemented and computational experiments were carried out; in the latter, the dimension of the communication network was chosen high enough to guarantee the legality of using the selected variants of algorithms for real communication networks. A representative series of experiments has shown that it is more expedient to use variants of the greedy heuristic algorithm for the group of problems under consideration, this paper contains a detailed argumentation of the result obtained. The considered problem can also be transferred to cases of optimal placement of other resources on the objects of the communication network. At the same time, if there is a specific objective function, there will be another formalization of restrictions. But the approaches to choosing the optimum may be identical, which makes the results obtained in the paper more important due to their participation in potential generalizations of the problem.

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Section: The Formal Graph Modelmentioning

confidence: 99%

Greedy and branches-and-boundaries methods for the optimal choice of a subset of vertices in a large communication network

Melnikov

Terentyeva²

2023

CAP

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“…However, the Dilworth number until now seems unexplored in a parameterized complexity context. For example, neither Gera et al [34] nor Sorge and Weller [50] list it. This is surprising, since the Dilworth number is bounded from above by the neighborhood diversity (see Section 2.3), which is a well-established parameter in parameterized complexity studies [4,[31][32][33]39], and it seems a logical step to analyze which parameterized algorithms for the neighborhood diversity can be strengthened to use the Dilworth number instead.…”

Section: Problem 11 (Multiple Hitting Set)mentioning

confidence: 99%

Serial and parallel kernelization of Multiple Hitting Set parameterized by the Dilworth number, implemented on the GPU

Bevern¹,

Kirilin²,

Skachkov³

et al. 2021

Preprint

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The NP-hard Multiple Hitting Set problem is finding a minimum-cardinality set intersecting each of the sets in a given input collection a given number of times. Generalizing a well-known data reduction algorithm due to Weihe, we show a problem kernel for Multiple Hitting Set parameterized by the Dilworth number, a graph parameter introduced by Foldes and Hammer in 1978 yet seemingly so far unexplored in the context of parameterized complexity theory. Using matrix multiplication, we speed up the algorithm to quadratic sequential time and logarithmic parallel time. We experimentally evaluate our algorithms. By implementing our algorithm on GPUs, we show the feasibility of realizing kernelization algorithms on SIMD (Single Instruction, Multiple Date) architectures.

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“…Besides these, we are not aware of any other results on exact (k, d)-colorings. Furthermore, Gera et al [14] drew attention to the problem by writing that it is one of six vertex-partitioning problems that has "received little or no attention".…”

Section: Introductionmentioning

confidence: 99%

Exact defective colorings of graphs

Cumberbatch¹,

Lauri²,

Mitillos³

2021

Preprint

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An exact (k, d)-coloring of a graph G is a coloring of its vertices with k colors such that each vertex v is adjacent to exactly d vertices having the same color as v. The exact d-defective chromatic number, denoted χ = d (G), is the minimum k such that there exists an exact (k, d)-coloring of G. In an exact (k, d)-coloring, which for d = 0 corresponds to a proper coloring, each color class induces a d-regular subgraph. We give basic properties for the parameter and determine its exact value for cycles, trees, and complete graphs. In addition, we establish bounds on χ = d (G) for all relevant values of d when G is planar, chordal, or has bounded treewidth. We also give polynomial-time algorithms for finding certain types of exact (k, d)-colorings in cactus graphs and block graphs. Our main result is on the computational complexity of d-Exact Defective k-Coloring in which we are given a graph G and asked to decide whether χ = d (G) ≤ k. Specifically, we prove that the problem is NP-complete for all d ≥ 1 and k ≥ 2.

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An Annotated Glossary of Graph Theory Parameters, with Conjectures

Cited by 9 publications

References 395 publications

Greedy and branches-and-boundaries methods for the optimal choice of a subset of vertices in a large communication network

Greedy and branches-and-boundaries methods for the optimal choice of a subset of vertices in a large communication network

Serial and parallel kernelization of Multiple Hitting Set parameterized by the Dilworth number, implemented on the GPU

Exact defective colorings of graphs

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