scholarly journals Equitable and semi-equitable coloring of cubic graphs and its application in batch scheduling

2015 ◽  
Vol 25 (1) ◽  
pp. 109-116 ◽  
Author(s):  
Hanna Furmańczyk ◽  
Marek Kubale
Keyword(s):  
Broad Spectrum ◽  
Unit Length ◽  
Batch Scheduling ◽  
Cubic Graphs ◽  
Equitable Coloring ◽  
Np Complete ◽  
Graph Parameters ◽  
Incompatibility Graph ◽  
Spectrum Of Graph ◽  
Uniform Processors

Abstract In the paper we consider the problems of equitable and semi-equitable coloring of vertices of cubic graphs. We show that in contrast to the equitable coloring, which is easy, the problem of semi-equitable coloring is NP-complete within a broad spectrum of graph parameters. This affects the complexity of batch scheduling of unit-length jobs with cubic incompatibility graph on three uniform processors to minimize the makespan.

scholarly journals Detection number of bipartite graphs and cubic graphs

2014 ◽  
Vol Vol. 16 no. 3 ◽  
Author(s):  
Frederic Havet ◽  
Nagarajan Paramaguru ◽  
Rathinaswamy Sampathkumar
Keyword(s):  
Positive Integer ◽  
Bipartite Graph ◽  
Connected Graph ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
International Audience ◽  
Np Complete

International audience For a connected graph G of order |V(G)| ≥3 and a k-labelling c : E(G) →{1,2,…,k} of the edges of G, the code of a vertex v of G is the ordered k-tuple (ℓ1,ℓ2,…,ℓk), where ℓi is the number of edges incident with v that are labelled i. The k-labelling c is detectable if every two adjacent vertices of G have distinct codes. The minimum positive integer k for which G has a detectable k-labelling is the detection number det(G) of G. In this paper, we show that it is NP-complete to decide if the detection number of a cubic graph is 2. We also show that the detection number of every bipartite graph of minimum degree at least 3 is at most 2. Finally, we give some sufficient condition for a cubic graph to have detection number 3.

scholarly journals Stable Divisorial Gonality is in NP

2020 ◽  
Author(s):  
Hans L. Bodlaender ◽  
Marieke van der Wegen ◽  
Tom C. van der Zanden
Keyword(s):  
Linear Programming ◽  
Algebraic Geometry ◽  
Integer Linear Programming ◽  
Connected Graph ◽  
Np Hard ◽  
Chip Firing ◽  
Np Complete ◽  
Graph Parameters

AbstractDivisorial gonality and stable divisorial gonality are graph parameters, which have an origin in algebraic geometry. Divisorial gonality of a connected graph G can be defined with help of a chip firing game on G. The stable divisorial gonality of G is the minimum divisorial gonality over all subdivisions of edges of G. In this paper we prove that deciding whether a given connected graph has stable divisorial gonality at most a given integer k belongs to the class NP. Combined with the result that (stable) divisorial gonality is NP-hard by Gijswijt et al., we obtain that stable divisorial gonality is NP-complete. The proof consists of a partial certificate that can be verified by solving an Integer Linear Programming instance. As a corollary, we have that the total number of subdivisions needed for minimum stable divisorial gonality of a graph with m edges is bounded by mO(mn).

scholarly journals Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines

2017 ◽  
Vol 65 (1) ◽  
pp. 29-34 ◽  
Author(s):  
H. Furmańczyk ◽  
M. Kubale
Keyword(s):  
Approximation Algorithm ◽  
General Problem ◽  
Unit Length ◽  
Mutual Exclusion ◽  
Np Hard ◽  
Uniform Machines ◽  
Identical Jobs ◽  
Incompatibility Graph

Abstract In the paper we consider the problem of scheduling n identical jobs on 4 uniform machines with speeds s1 ≥ s2 ≥ s3 ≥ s4, respectively. Our aim is to find a schedule with a minimum possible length. We assume that jobs are subject to some kind of mutual exclusion constraints modeled by a bipartite incompatibility graph of degree Δ, where two incompatible jobs cannot be processed on the same machine. We show that the general problem is NP-hard even if s1 = s2 = s3. If, however, Δ ≤ 4 and s1 ≥ 12s2, s2 = s3 = s4, then the problem can be solved to optimality in time O(n1.5). The same algorithm returns a solution of value at most 2 times optimal provided that s1 ≥ 2s2. Finally, we study the case s1 ≥ s2 ≥ s3 = s4 and give a 32/15-approximation algorithm running also in O(n1.5) time.

scholarly journals Equitable Partition of Graphs into Independent Sets and Cliques

2019 ◽  
Author(s):  
Bruno Monteiro ◽  
Vinicius Dos Santos
Keyword(s):  
Polynomial Time ◽  
Independent Sets ◽  
Equitable Partition ◽  
Equitable Coloring ◽  
Vertex Set ◽  
Np Complete

A graph is (k, l) if its vertex set can be partitioned into k independent sets and l cliques. Deciding if a graph is (k, l) can be seen as a generalization of coloring, since deciding is a graph belongs to (k, 0) corresponds to deciding if a graph is k-colorable. A coloring is equitable if the cardinalities of the color classes differ by at most 1. In this paper, we generalize both the (k, l) and the equitable coloring problems, by showing that deciding whether a given graph can be equitably partitioned into k independent sets and l cliques is solvable in polynomial time if max(k, l) 2, and NP complete otherwise.

scholarly journals On Independent Secondary Dominating Sets in Generalized Graph Products

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2399
Author(s):  
Adrian Michalski ◽  
Paweł Bednarz
Keyword(s):  
Complete Characterization ◽  
Asymmetric Structure ◽  
Graph Products ◽  
Dominating Sets ◽  
Complete Problem ◽  
Join Of Graphs ◽  
Np Complete ◽  
Graph Parameters ◽  
Domination In Graphs

In 2008, Hedetniemi et al. introduced (1,k)-domination in graphs. The research on this concept was extended to the problem of existence of independent (1,k)-dominating sets, which is an NP-complete problem. In this paper, we consider independent (1,1)- and (1,2)-dominating sets, which we name as (1,1)-kernels and (1,2)-kernels, respectively. We obtain a complete characterization of generalized corona of graphs and G-join of graphs, which have such kernels. Moreover, we determine some graph parameters related to these sets, such as the number and the cardinality. In general, graph products considered in this paper have an asymmetric structure, contrary to other many well-known graph products (Cartesian, tensor, strong).

Metallurgical Effects of Grain Boundary Ledges

1977 ◽  
Vol 35 ◽  
pp. 126-129
Author(s):  
L.E. Murr
Keyword(s):  
Grain Boundary ◽  
Quantitative Data ◽  
Mechanical Treatment ◽  
Unit Length ◽  
Physical And Mechanical Properties ◽  
Direct Observations ◽  
Transmission Electron ◽  
Properties Of Materials ◽  
Thermo Mechanical Treatment ◽  
Ledge Density

Ledges in grain boundaries can be identified by their characteristic contrast features (straight, black-white lines) distinct from those of lattice dislocations, for example1,2 [see Fig. 1(a) and (b)]. Simple contrast rules as pointed out by Murr and Venkatesh2, can be established so that ledges may be recognized with come confidence, and the number of ledges per unit length of grain boundary (referred to as the ledge density, m) measured by direct observations in the transmission electron microscope. Such measurements can then give rise to quantitative data which can be used to provide evidence for the influence of ledges on the physical and mechanical properties of materials.It has been shown that ledge density can be systematically altered in some metals by thermo-mechanical treatment3,4.

Characterization of microtubules by dark-field STEM and replication

1991 ◽  
Vol 49 ◽  
pp. 152-153
Author(s):  
S.B. Andrews ◽  
R.D. Leapman ◽  
P.E. Gallant ◽  
T.S. Reese
Keyword(s):  
Protein Interactions ◽  
Low Dose ◽  
Unit Length ◽  
Dark Field ◽  
Carbon Films ◽  
Microtubule Associated Proteins ◽  
Molecular Weights ◽  
Freeze Dried ◽  
Protein Motors ◽  
Associated Proteins

As part of a study on protein interactions involved in microtubule (MT)-based transport, we used the VG HB501 field-emission STEM to obtain low-dose dark-field mass maps of isolated, taxol-stabilized MTs and correlated these micrographs with detailed stereo images from replicas of the same MTs. This approach promises to be useful for determining how protein motors interact with MTs. MTs prepared from bovine and squid brain tubulin were purified and free from microtubule-associated proteins (MAPs). These MTs (0.1-1 mg/ml tubulin) were adsorbed to 3-nm evaporated carbon films supported over Formvar nets on 600-m copper grids. Following adsorption, the grids were washed twice in buffer and then in either distilled water or in isotonic or hypotonic ammonium acetate, blotted, and plunge-frozen in ethane/propane cryogen (ca. -185 C). After cryotransfer into the STEM, specimens were freeze-dried and recooled to ca.-160 C for low-dose (<3000 e/nm2) dark-field mapping. The molecular weights per unit length of MT were determined relative to tobacco mosaic virus standards from elastic scattering intensities. Parallel grids were freeze-dried and rotary shadowed with Pt/C at 14°.

A Strengths-Based Approach to Autism: Neurodiversity and Partnering With the Autism Community

2017 ◽  
Vol 2 (1) ◽  
pp. 56-68 ◽  
Author(s):  
Amy L. Donaldson ◽  
Karen Krejcha ◽  
Andy McMillin
Keyword(s):  
Broad Spectrum ◽  
First Language ◽  
Family Members ◽  
Community Identity ◽  
Speech Language Pathologists

The autism community represents a broad spectrum of individuals, including those experiencing autism, their parents and/or caregivers, friends and family members, professionals serving these individuals, and other allies and advocates. Beliefs, experiences, and values across the community can be quite varied. As such, it is important for the professionals serving the autism community to be well-informed about current discussions occurring within the community related to neurodiversity, a strengths-based approach to partnering with autism community, identity-first language, and concepts such as presumed competence. Given the frequency with which speech-language pathologists (SLPs) serve the autism community, the aim of this article is to introduce and briefly discuss these topics.

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