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Universal maximal packing functions of graphs

E.J. Cockayne, O. Favaron, C.M. Mynhardt
1996 Discrete Mathematics  
We consider the existence of universal maximal packing functions, i.e. maximal packing functions (MPFs) whose convex combinations with all other MPFs are themselves MPFs.  ...  A packing function of a graph G = (V, E) is a function f: V --* [0, 1] such that for each v ~ V, denotes the closed neighbourhood of vertex v).  ...  The work was completed while the first author was enjoying the hospitality of the University of South Africa.  ... 
doi:10.1016/0012-365x(95)00097-g fatcat:fm7t4mxlzfgwdf7vyvwqnddahe

Page 51 of Mathematical Reviews Vol. , Issue 98A [page]

1998 Mathematical Reviews  
We consider the existence of universal maximal packing functions, i.e. maximal packing functions (MPFs) whose convex combinations with all other MPFs are themselves MPFs.”  ...  (SA-SAFR-A; Pretoria) Universal maximal packing functions of graphs. (English summary) Discrete Math. 159 (1996), no. 1-3, 57-68.  ... 

Optimal Information Rate of Secret Sharing Schemes on Trees

László Csirmaz, Gábor Tardos
2013 IEEE Transactions on Information Theory  
Secret sharing scheme; information rate; graph; fractional packing and cover; entropy method.  ...  Canada Research Chair of Computational and Discrete Geometry at the Simon Fraser University, BC, Canada, since 2005.  ...  a maximal weight function.  ... 
doi:10.1109/tit.2012.2236958 fatcat:cvlehsqmgrbybofdm3dbec4kue

Optimal information rate of secret sharing schemes on trees [article]

L. Csirmaz, G. Tardos
2013 arXiv   pre-print
Our result follows from a lower and an upper bound on the information rate that applies for any graph and happen to coincide for trees because of a correspondence between the size of the largest core and  ...  We determine this value for all trees: it is 1/(2-1/c), where c is the size of the largest core of the tree.  ...  a maximal weight function.  ... 
arXiv:1302.4609v1 fatcat:is4p7zlnxjhsvfukejj3aswxje

Mining protein family specific residue packing patterns from protein structure graphs

Jun Huan, Wei Wang, Deepak Bandyopadhyay, Jack Snoeyink, Jan Prins, Alexander Tropsha
2004 Proceedings of the eighth annual international conference on Computational molecular biology - RECOMB '04  
Avg # of edges per residue Edge length prune (Angstrom) Avg vertex degree of graph as function of edge length 12/29/2005 7 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Graph Database Mining -Problem  ...  (CAM) of a graph G is the maximal adjacency matrix for G under a total ordering defined on adjacency matrices.  ...  A group of family specific motifs are identified and some of them correlate well with the know function of group of proteins studied.  ... 
doi:10.1145/974614.974655 dblp:conf/recomb/HuanWBSPT04 fatcat:mx3mx33fobcwxitykeh4qj3yme

A nonsmooth program for jamming hard spheres [article]

Peter Hinow
2012 arXiv   pre-print
We propose an algorithm for the maximization of this maximal radius function and present examples for 5 - 8 disks in the unit square and 4 - 6 spheres in the unit cube.  ...  We present a nonsmooth function whose local extrema are the radii of jammed packings (where no subset of spheres can be moved keeping all others fixed) and show that for a fixed number of spheres there  ...  Boris Okun and Michael Hero (Department of Mathematical Sciences, University of Wisconsin -Milwaukee) and Michael Monagan (Department of Mathematics, Simon Fraser University, Burnaby, British Columbia)  ... 
arXiv:1209.4053v1 fatcat:rkgjpatv5fd7xi6q45g6h3l5ku

Fractional edge domination in graphs

S. Arumugam, Jerry Sithara
2009 Applicable Analysis and Discrete Mathematics  
The authors are thankful to the Department of Science and Technology, New Delhi for its support through the Project SR/S4/MS/282/05.  ...  p f (G) = min{|g| : g is a maximal edge packing function of G} and P f (G) = max{|g| : g is a maximal edge packing function of G}. Observation 2.5.  ...  Characterize graphs G for which the space of all maximal edge irredundant functions is compact. Problem 3.5.  ... 
doi:10.2298/aadm0902359a fatcat:gn7lamz4jvgdbhgzexdeivwaru

Faster Fixed-Parameter Tractable Algorithms for Matching and Packing Problems

M. R. Fellows, C. Knauer, N. Nishimura, P. Ragde, F. Rosamond, U. Stege, D. M. Thilikos, S. Whitesides
2007 Algorithmica  
We obtain faster algorithms for problems such as r-dimensional matching, r-set packing, graph packing, and graph edge packing when the size k of the solution is considered a parameter.  ...  We first establish a general framework for finding and exploiting small problem kernels (of size polynomial in k). Previously such a kernel was known only for triangle packing.  ...  We then have the following lemma, analogous to Lemma 5: Lemma 6 For the problems r-Set Packing, Graph Packing, and Graph Edge-Packing, if S ⊆ A and K is the output of the function Kernel-Construct(S),  ... 
doi:10.1007/s00453-007-9146-y fatcat:kxt6holtirftfccq46oluw2yc4

Faster Fixed-Parameter Tractable Algorithms for Matching and Packing Problems [chapter]

Michael R. Fellows, C. Knauer, N. Nishimura, P. Ragde, F. Rosamond, U. Stege, Dimitrios M. Thilikos, S. Whitesides
2004 Lecture Notes in Computer Science  
We obtain faster algorithms for problems such as r-dimensional matching, r-set packing, graph packing, and graph edge packing when the size k of the solution is considered a parameter.  ...  We first establish a general framework for finding and exploiting small problem kernels (of size polynomial in k). Previously such a kernel was known only for triangle packing.  ...  We then have the following lemma, analogous to Lemma 5: Lemma 6 For the problems r-Set Packing, Graph Packing, and Graph Edge-Packing, if S ⊆ A and K is the output of the function Kernel-Construct(S),  ... 
doi:10.1007/978-3-540-30140-0_29 fatcat:vg5dk5mdvrfxvnycgjfszlfjg4

Arbitrary Overlap Constraints in Graph Packing Problems [article]

Alejandro López-Ortiz, Jazmín Romero
2016 arXiv   pre-print
Let V^r be all possible subsets of vertices of V(G) each of size at most r, and α: V^r ×V^r →{0,1} be a function.  ...  The Π-Packing with α()-Overlap problem seeks at least k induced subgraphs in a graph G subject to: (i) each subgraph has at most r vertices and obeys a property Π, and (ii) for any pair H_i,H_j, with i  ...  In this work, we introduce the r-Set Packing and Π-Set Packing with α()-Overlap problems as more universal versions for the problem of packing graphs and sets subject to overlap constraints modeled by  ... 
arXiv:1601.03676v1 fatcat:o3bp2ri7tvfermuuxwynljmria

On packing bipartite graphs

P�ter Hajnal, M�ri� Szegedy
1992 Combinatorica  
A theorem of Catlin, Spencer and Sauer gives a su cient condition for the existence of packing in terms of the product of the maximal degrees of G and H. We improve this theorem for bipartite graphs.  ...  G and H, two simple graphs, can be packed if G is isomorphic to a subgraph of H, the complement of H.  ...  The following few theorems give su cient conditions on the number of edges for the existence of a packing. (ii) If G; H 2 B u;w and jE(G)jjE(H)j < uw, then G and H can be packed as bipartite graphs.  ... 
doi:10.1007/bf01285818 fatcat:ibasee5v4jdybaoyqyv2bawpym

Packing functions and graphs with perfect closed neighbourhood matrices [article]

Mariana Escalante, Erica Hinrichsen, Valeria. Leoni
2018 arXiv   pre-print
In this work we consider a straightforward linear programming formulation of the recently introduced {k}-packing function problem in graphs, for each fixed value of the positive integer number k.  ...  We present a necessary and sufficient condition for a graph to have a clique-node closed neighbourhood matrix. Finally, we study the perfection of the graph of maximal cliques associated to N[G].  ...  Limited packing functions For a graph G = (V, E) and a positive integer k, a function f : V → Z + is a {k}-packing function of G [9] , if for each node v ∈ V , f (N [v]) = w∈N [v] f (w) ≤ k.  ... 
arXiv:1812.09422v1 fatcat:clap6u63gjbrzjqdh6b2bgwty4

Covering and Packing in Linear Space [chapter]

Andreas Björklund, Thore Husfeldt, Petteri Kaski, Mikko Koivisto
2010 Lecture Notes in Computer Science  
Given a family of subsets of an n-element universe, the kcover problem asks whether there are k sets in the family whose union contains the universe; in the k-packing problem the sets are required to be  ...  Thus, for instance, the chromatic or domatic number of an n-vertex graph can be found in time within a polynomial factor of 2 n and space proportional to the number of maximal independent sets, O(1.442  ...  This bottleneck can be removed by taking a more balanced scheme with n 1 and n 2 about equal, yielding time and space O * (2 n/2 ) for each of the O * (2 n/2 ) processors.  ... 
doi:10.1007/978-3-642-14165-2_61 fatcat:udxf6oa5onclxk5p6yyrygxnxy

Covering and packing in linear space

Andreas Björklund, Thore Husfeldt, Petteri Kaski, Mikko Koivisto
2011 Information Processing Letters  
Given a family of subsets of an n-element universe, the kcover problem asks whether there are k sets in the family whose union contains the universe; in the k-packing problem the sets are required to be  ...  Thus, for instance, the chromatic or domatic number of an n-vertex graph can be found in time within a polynomial factor of 2 n and space proportional to the number of maximal independent sets, O(1.442  ...  This bottleneck can be removed by taking a more balanced scheme with n 1 and n 2 about equal, yielding time and space O * (2 n/2 ) for each of the O * (2 n/2 ) processors.  ... 
doi:10.1016/j.ipl.2011.08.002 fatcat:rj32rkvxknefnerpdocpregzy4

Optimal packings of up to six equal circles on a triangular flat torus

William Dickinson, Daniel Guillot, Anna Keaton, Sandi Xhumari
2011 Journal of Geometry  
The unique globally maximally dense packing of n − 1 circles on the standard triangular torus is congruent to C with one circle removed.  ...  Let n > 1 be a triangular packing number so that n − 1 is not a triangular packing number and C be the packing of n circles on the standard triangular torus that lift to the triangular close packing in  ...  Some packings are local maximums of the density function.  ... 
doi:10.1007/s00022-011-0099-6 fatcat:stk5miywnjdgrfbd22s7vz72iq
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