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On Cyclic Packing of a Tree

Stephan Brandt, Mariusz Woźniak
2004 Graphs and Combinatorics  
We prove that there exists a packing of bn=2c copies of a tree of size dn=2e into K n . Moreover, the proof provides an easy algorithm.  ...  We use the following terminology: A packing of k copies of a graph G will be called a cyclic packing of G if there exists a permutation r on V ðGÞ such that the graphs G, rðGÞ, r 2 ðGÞ,..., r kÀ1 ðGÞ are  ...  Next, because of the methods we use, we consider the case of the packing of k copies of a tree. On the other hand we obtain something more than the existence of the packing.  ... 
doi:10.1007/s00373-004-0583-y fatcat:gtfakzbqgbcuzfzswp255uirfy

Page 69 of Mathematical Reviews Vol. , Issue 90A [page]

1990 Mathematical Reviews  
Finally, there is a report on some special packings: a tree with an arbitrary graph, a tree with an (n,n — 1) graph, a tree with an (n, n) graph, two (n,n —1) graphs, and two graphs of order n.  ...  In presenting this survey, the author states that he will concen- trate on two packing problems: one concerning dense packings of trees of different sizes in K, and the other that of packing two graphs  ... 

PLANAR PACKING OF CYCLES AND UNICYCLIC GRAPHS

Agnieszka Gôrlich
2009 Demonstratio Mathematica  
In 2002 Garcia et al. have shown that a non-star tree T is planary packable into a complete graph K n .  ...  In this paper we show that for any packable cycle C n except of the case n = 5 and n = 7 there exists a planar packing into K n . We also generalize this result to certain classes of unicyclic graphs.  ...  A cyclic packing of G is a cyclic permutation a : V(G) -> V(G) (so a has exactly one cycle in its decomposition into cycles). A basic result concerning packing problem [1] is: THEOREM 1.  ... 
doi:10.1515/dema-2009-0402 fatcat:ptqhl4plerg7db7fm77o3azyb4

3-D floorplanning using labeled tree and dual sequences

Renshen Wang, Evangeline F. Y. Young, Yi Zhu, Fan Chung Graham, Ronald Graham, Chung-Kuan Cheng
2008 Proceedings of the 2008 international symposium on Physical design - ISPD '08  
We present a complete representation for general non-slicing 3-D floorplan or packing structures, which uses a labeled tree and dual sequences.  ...  For each compact placement, there is a corresponding encoding. The number of possible tree-sequence combinations is (n + 1) n−1 (n!) 2 , the lowest among complete 3-D representations up to date.  ...  Since the cyclic relation problem is resolved, the blocks can be packed in a sequential order like in 2-D packing: Push the blocks one by one, each to a corner until the block cannot be pushed further  ... 
doi:10.1145/1353629.1353641 dblp:conf/ispd/WangYZGGC08 fatcat:j3kqlepacfagxdbvlvncibku5i

Planar packing of cycles and unicyclic graphs

Agnieszka Görlich
2009 Demonstratio Mathematica  
AbstractWe say that a graphIn this paper we show that for any packable cycle  ...  A cyclic packing of G is a cyclic permutation σ : V (G) → V (G) (so σ has exactly one cycle in its decomposition into cycles).  ...  Identifying a cycle segment with the equator of a sphere and drawing edges of G on the northern hemisphere and edges of σ(G) on the southern hemisphere one can see that σ is a planar cyclic packing.  ... 
doi:10.1515/dema-2013-0193 fatcat:le7t2lngyven3ooy3ffranggj4

A note on uniquely embeddable forests

Justyna Otfinowska, Mariusz Woźniak
2013 Discussiones Mathematicae Graph Theory  
Let F be a forest of order n. It is well known that if F = S n , a star of order n, then there exists an embedding of F into its complement F .  ...  In this note we consider a problem concerning the uniqueness of such an embedding.  ...  It is easy to see that the tree T ′ consists of the path joining a and b and (maybe) of some vertices Since T ′ contains a path on five vertices, T ′ is not a star and by Theorem 2 there is a cyclic  ... 
doi:10.7151/dmgt.1651 fatcat:pxqaumwspfhwhiab2t4v53vuou

Page 648 of Mathematical Reviews Vol. , Issue 95b [page]

1995 Mathematical Reviews  
The algorithm is based on the fact that in each tree T there exists a path a such that P(T) = P(T —a)+|E|.  ...  A pentagonal packing PP(n;t) is a family of ¢ edge-disjoint pen- tagons in the complete graph K,,. A pentagonal packing is maximal if the complement of the union of its pentagons is pentagon-free.  ... 

CICLIC: A Tool to Generate Feasible Cyclic Schedules

J. Yepez, J. Guardia, M. Velasco, J. Ayza, R. Castane, P. Marti, J.M. Fuertes
2006 2006 IEEE International Workshop on Factory Communication Systems  
A common way of implementing periodic real-time systems is by means of a cyclic scheduler.  ...  In this paper, a tool having this function, and named CICLIC is described. The design mechanism is based on algorithms of exhaustive search that use heuristic rules in optimizing searching paths.  ...  ACKNOWLEDGEMENTS The research for this paper has been sponsored by the Ministry of Science and Technology, through reference project No. DPI20002/01621.  ... 
doi:10.1109/wfcs.2006.1704187 fatcat:ypjlk26ckjezdagizmmmnd5nca

Page 1864 of Mathematical Reviews Vol. , Issue 94d [page]

1994 Mathematical Reviews  
It follows that the Gyarfas tree packing conjecture and Fishburn’s conjectures hold for trees belonging to these families.  ...  The graph G is cyclically m-edge- connected if the size of every cyclic edge cutset is m or greater.  ... 

Edge-disjoint packings of graphs

Derek G. Corneil, Shigeru Masuyama, S. Louis Hakimi
1994 Discrete Applied Mathematics  
In this paper we study two types of edge-disjoint packings of graphs.  ...  The second edge-disjoint packing problem concerns partial subgraphs and asks whether a given graph H contains at least k copies G as partial subgraphs such that no two such copies of G share an edge.  ...  F-G'lP packing problems are also of interest when various restrictions such as planarity or being a tree are placed on the host graphs.  ... 
doi:10.1016/0166-218x(92)00153-d fatcat:xjlp5zevqfb5dkq4d62it7f6ja

On cyclically embeddable graphs

Mariusz Woźniak
1999 Discussiones Mathematicae Graph Theory  
An embedding of a simple graph G into its complement G is a permutation σ on V (G) such that if an edge xy belongs to E(G), then σ(x)σ(y) does not belong to E(G).  ...  In this note we consider some families of embeddable graphs such that the corresponding permutation is cyclic.  ...  However, if we assume in addition that G is a tree, we have the following result (cf. [11] ). Theorem 4. Let T be a tree of order n. If T = S n then there exists a cyclic embedding of G.  ... 
doi:10.7151/dmgt.1099 fatcat:q3jrvxd6vvhpbfarncjxnwv2i4

Page 4118 of Mathematical Reviews Vol. , Issue 98G [page]

1998 Mathematical Reviews  
And it is trees that this paper focuses on; in particular characterizing all those trees JT of diameter 3 for which there is a cyclic orthogonal double cover of K, by T.  ...  Finally it is shown that for all but 13 of the 2087 trees T on n = 11, 12 or 13 vertices, there is a cyclic orthogonal double cover of K, by 7.  ... 

Page 7188 of Mathematical Reviews Vol. , Issue 95m [page]

1995 Mathematical Reviews  
The size of the largest node in any 1|-hinge tree is shown to be an invariant of the hypergraph, called the de- gree of cyclicity (acyclic hypergraphs have the degree of cyclicity 2).  ...  A k-hinge tree is a set of minimal k-hinges that cover all edges of H and form a tree with respect to intersection.  ... 

A Round Trip Pattern for Building Decision Trees

Liang Jia, Liuhong Yan
2016 International Journal of Database Theory and Application  
This cyclic pattern starts at creating a decision tree from initial table altered according to Bayes theorem combined with Pareto values of objective functions.  ...  This paper introduces a recursive procedure named round trip pattern for generating decision trees based on alterable decision tables.  ...  The overall number of processed packs is shown in Figure 6 . The packs are cyclically classified at start of each month and the number is increasing as time elapsed.  ... 
doi:10.14257/ijdta.2016.9.7.16 fatcat:vg4ofrvzvbbavdv6r2p6usvpgm

On cyclically embeddable (n,n−1)-graphs

Mariusz Woźniak
2002 Discrete Mathematics  
An embedding of a simple graph G into its complement G is a permutation on V (G) such that if an edge xy belongs to E(G), then (x) (y) does not belong to E(G).  ...  We prove that with few exceptions the corresponding permutation may be chosen as cyclic one.  ...  Then there exists a cyclic embedding of G. The general references for these and other packing problems are [1, 7, 5] (see also [8] ).  ... 
doi:10.1016/s0012-365x(01)00339-9 fatcat:n3pdowolqja4raopwd432wl4hm
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