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Enumerating minimal dominating sets in the (in)comparability graphs of bounded dimension posets
[article]

2020
*
arXiv
*
pre-print

., for

arXiv:2004.07214v1
fatcat:5atat6z2hrbzxci2iv7kbumrfa
*comparability**graphs**of**bounded**dimension**posets*. ... It can be reduced to*enumerating**minimal**dominating**sets**in*a*graph*,*in*fact even to*enumerating**minimal**dominating**sets**in*an incomparability*graph*. ... Discussions We provided an incremental-polynomial (resp. polynomial-delay) algorithm*enumerating**the**minimal**dominating**sets**in**the**comparability*(resp. incomparability)*graphs**of**bounded**dimension**posets*...##
###
Maximal Antichain Lattice Algorithms for Distributed Computations
[chapter]

2013
*
Lecture Notes in Computer Science
*

Since

doi:10.1007/978-3-642-35668-1_17
fatcat:72f5jpjdfbg75em3glzoloa4ea
*the*lattice may be exponential*in**the*size*of**the*computation, we get a significant reduction*in**the*space complexity. ... For*the*purposes*of*analyzing offline traces, we also propose new*enumeration*algorithms to traverse*the*lattice. ... This claim follows from*the*facts that A ⊆*minimal*(I c ), D(A) = I and U [*minimal*(I c )] = I c .*The*time complexity*of*ILMA is*dominated*by Step 2. ...##
###
On H-Topological Intersection Graphs
[chapter]

2017
*
Lecture Notes in Computer Science
*

*The*minimum vertex leafage problem can be solved

*in*n O( ) -time via a somewhat elaborate

*enumeration*

*of*

*minimal*1 tree representations

*of*G with exactly leaves where is

*the*leafage

*of*G [6]. ... Our paper is

*the*first study

*of*

*the*recognition and

*dominating*

*set*problems

*of*this large collection

*of*intersection classes

*of*

*graphs*. ...

*In*particular,

*the*algorithm runs

*in*n O( H ) time. E

*Bounded*List Coloring

*of*Co-

*comparability*

*Graphs*We show how to solve

*the*problem

*of*

*bounded*list coloring

*of*co-

*comparability*

*graphs*. ...

##
###
Enumeration of Enumeration Algorithms
[article]

2016
*
arXiv
*
pre-print

*In*this paper, we

*enumerate*

*enumeration*problems and algorithms. This survey is under construction. If you know some results not

*in*this survey or there is anything wrong, please let me know. ... Reference [254] 3.2 Independent

*set*3.2.1

*Enumeration*

*of*all maximal independent

*set*

*of*a hypergraph

*of*

*bounded*

*dimension*Input A hypergraph H

*of*

*bounded*

*dimension*. ... Reference [247] 2.12

*Dominating*

*set*2.12.1

*Enumeration*

*of*all

*minimal*

*dominating*

*sets*

*in*a

*graph*Input A

*graph*G = (V, E). Output All

*minimal*

*dominating*

*sets*

*in*G. ...

##
###
On Finding Minimal Infrequent Elements in Multi-dimensional Data Defined over Partially Ordered Sets
[article]

2014
*
arXiv
*
pre-print

We study

arXiv:1411.2275v1
fatcat:nhft764wevcjra4jevbpf3hrwe
*the*complexity*of*finding all*minimal*infrequent elements for some interesting classes*of**posets*. ... We consider databases*in*which each attribute takes values from a partially ordered*set*(*poset*). ...*The*precedence*graph**of*a*poset*P is a directed acyclic*graph*with vertex*set*P, and*set**of*arcs {(x, y) : y is an immediate successor*of*x}. ...##
###
Minimizing Costs of Resource Requirements in Project Networks Subject to a Fixed Completion Time

1984
*
Operations Research
*

It is even possible to characterize all smallest

doi:10.1287/opre.32.1.89
fatcat:h7wkcbbj4bejljnnd3fnnxvo7u
*sets*having this property (so-called*minimal**dominating*systems). This is done*in*Section 2. ... These methods are based on algorithms recently developed*in**graph*theory for*comparability**graph*and interval*graph*recognition and orientation.*The*last two parts are self-contained. ... ACKNOWLEDGMENTS This work was supported*in*part by*the*Minister fur Wissenschaft und Forschung des Landes Nordrhein-Westfalen. ...##
###
The Order Dimension of Planar Maps Revisited

2014
*
SIAM Journal on Discrete Mathematics
*

It states that

doi:10.1137/130945284
fatcat:bwbqjz2cbbfx7chitrqtkymeai
*the*order*dimension**of**the*incidence*poset*P M*of*vertices, edges and faces*of*a planar map M has*dimension*at most 4. ... Schnyder characterized planar*graphs**in*terms*of*order*dimension*. This seminal result found several extensions. ...*In*Q G*the**set*X is*the**set**of**minimal*elements and Y is*the**set**of*maximal elements. ...##
###
Cardinal invariants of closed graphs
[article]

2020
*
arXiv
*
pre-print

*In*particular, we address

*the*question when it is consistent for

*the*

*bounding*number to be strictly smaller than

*the*smallest size

*of*a

*set*not covered by countably many compact G-anticliques. ... We also provide a descriptive

*set*theoretic characterization

*of*

*the*class

*of*analytic

*graphs*with countable coloring number. ...

*The*metric spaces

*of*finite

*dimension*behave quite differently

*in*this respect than

*the*infinite

*dimensional*ones [4] .

*The*

*dimension*fault line appears

*in*

*the*following example: Example 5.6. ...

##
###
On H-Topological Intersection Graphs
[article]

2021
*
arXiv
*
pre-print

*The*algorithm for H-

*graphs*adapts to an XP-time algorithm for

*the*independent

*set*and

*the*independent

*dominating*

*set*problems on H-

*graphs*. ...

*In*fact, these results apply to classes

*of*

*graphs*with treewidth

*bounded*by a function

*of*

*the*clique number. ... By Theorem 19 a

*the*

*graph*G is k-thin. It can be tested

*in*O(n 3 ) time whether

*the*

*graph*G is s-colorable [4] . ...

##
###
A survey of the Shi arrangement
[article]

2020
*
arXiv
*
pre-print

Shi,

arXiv:1909.01257v3
fatcat:xpdn64ylbzfzvhmp7tmmkfeopa
*in*1986, proved part*of*this conjecture. As a byproduct, he introduced*the*Shi arrangement*of*hyperplanes and found a few*of*its remarkable properties. ...*The*Shi arrangement has since become a central object*in*algebraic combinatorics. ... See Athanasiadis [Ath05] or Haiman [Hai94] for more information. (4)*The**enumeration**of**bounded*regions has nice results, which we have not discussed. ...##
###
A central partition of molecular conformational space. I. Basic structures
[article]

2002
*
arXiv
*
pre-print

On

arXiv:physics/0204035v1
fatcat:473xsirlozcdpd4iw3tfi5mvvq
*the*basis*of*empirical evidence from molecular dynamics simulations, molecular conformational space can be described by means*of*a partition*of*central conical regions characterized by*the**dominance*... This work presents a geometric and combinatorial description*of*this structure. ... It can be shown that*the*partially ordered*set*(*poset*) L thus defined is a lattice, that is: for all pairs x , y ǫ L there is a least upper*bound*and a greatest lower*bound*. ...##
###
Master index of volumes 81–90

1991
*
Discrete Mathematics
*

*The*least point covering and

*domination*numbers

*of*a

*graph*Sanchis, L.A., Maximum number

*of*edges

*in*connected

*graphs*with a given

*domination*number Sato, I.,

*Enumeration*

*of*packed

*graphs*(Note) Savicky ... Lin, C.,

*The*

*dimension*

*of*

*the*Cartesian product

*of*

*posets*Lin, M.-C. and S. Lin, Codes with multi-level error-correcting capabilities Lin, S., see Lin, M.-C. Lindner, C.C., see Hoffman, D.G. ...

##
###
Homological algebra of modules over posets
[article]

2020
*
arXiv
*
pre-print

Tameness and its homological theory are new even

arXiv:2008.00063v2
fatcat:ze325f7r5jbyrp5zwdgrclg7ei
*in**the*finitely generated discrete*setting**of*N^n-gradings, where tame is materially weaker than noetherian. ... Homological algebra*of*modules over*posets*is developed, as closely parallel as possible to that*of*finitely generated modules over noetherian commutative rings,*in**the*direction*of*finite presentations ...*Comparing*Theorems 6.12 and 5.19, what happened to*minimality*? It is not clear*in*what generality*minimality*can be characterized. ...##
###
Efficient skyline evaluation over partially ordered domains

2010
*
Proceedings of the VLDB Endowment
*

Nevertheless these methods either use stronger notions

doi:10.14778/1920841.1920996
fatcat:ejxwsc27y5bs3jlqtp2d6ajpam
*of**dominance*, which generate false positives, or require expensive*dominance*checks. ...*The*first method uses an appropriate mapping*of*a partial order to a total order, inspired by*the*lattice theorem and an off-*the*-shelf skyline algorithm. ... This allows for efficient*dominance*checking between values*in**the*domain; otherwise, traversal*of**the**poset**graph*might be required. ...##
###
The Shi arrangement and the Ish arrangement
[article]

2010
*
arXiv
*
pre-print

Moreover, all

arXiv:1009.1655v2
fatcat:src3lz3nznh6bgznerwz2lagdi
*of*these results hold*in**the*greater generality*of*"deleted" Shi and Ish arrangements corresponding to an arbitrary subgraph*of**the*complete*graph*. ... For example,*the*Shi and Ish arrangements share*the*same characteristic polynomial,*the*same numbers*of*regions,*bounded*regions,*dominant*regions, regions with c "ceilings" and d "degrees*of*freedom", ... This project arose out*of*an AMS sectional meeting held at Penn State*in*October 2009. ...
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