Independent spanning trees of chordal rings

Iwasaki, Yukihiro; Kajiwara, Yuka; Obokata, Koji; Igarashi, Yoshihide

doi:10.1007/bfb0045110

Cited by 16 publications

(15 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It was conjectured that for any k-connected graph there exist k ISTs rooted at its any vertex [23]. The conjecture has been proved true for kconnected graphs with k 4 [3,4,11,23] and some classes of graphs or digraphs (in particular, interconnection networks) such as planar graphs [10], product graphs [15], chordal rings [12,19], star graphs [16], de Bruijn and Kautz 0020-0190/$ -see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2010.11.018 digraphs [6], recursive circulant graphs [20,21], hypercubes [22], locally twisted cubes [14] and torus [17], etc.…”

Section: J01354)mentioning

confidence: 99%

Parallel construction of optimal independent spanning trees on Cartesian product of complete graphs

Chen

2011

Information Processing Letters

View full text Add to dashboard Cite

Section: J01354)mentioning

confidence: 99%

Parallel construction of optimal independent spanning trees on Cartesian product of complete graphs

Chen

2011

Information Processing Letters

View full text Add to dashboard Cite

“…From then on, this conjecture has been shown to be true for k-connected graphs with k 4 (see [11,13,20,48] for k = 2, 3, 4, respectively) and is still open for k 5. Also, this conjecture has been confirmed for several restricted classes of graphs, e.g., graphs related to planarity [18,19,25,26], graphs defined by Cartesian product [6,27,29,30,34,40,44], variations of hypercubes [5,[8][9][10]24,32,33,38,49], special Cayley graphs [22,23,28,39,42,43], and chordal ring [21,41]. In particular, [5,[7][8][9][10][32][33][34]40,49] are published after 2012.…”

Section: Introductionmentioning

confidence: 87%

A fully parallelized scheme of constructing independent spanning trees on Möbius cubes

Yang

Chang

et al. 2014

J Supercomput

View full text Add to dashboard Cite

A set of spanning trees in a graph is said to be independent (ISTs for short) if all the trees are rooted at the same node r and for any other node v( = r ), the paths from v to r in any two trees are node-disjoint except the two end nodes v and r . It was conjectured that for any n-connected graph there exist n ISTs rooted at an arbitrary node. Let N = 2 n be the number of nodes in the n-dimensional Möbius cube M Q n . Recently, for constructing n ISTs rooted at an arbitrary node of M Q n , Cheng et al. (Comput J 56(11):1347(Comput J 56(11): -1362(Comput J 56(11): , 2013 and (J Supercomput 65 (3):1279-1301, 2013), respectively, proposed a sequential algorithm to run in O(N log N ) time and a parallel algorithm that takes O(N ) time using log N processors. However, the former algorithm is executed in a recursive fashion and thus is hard to be parallelized. Although the latter algorithm can simultaneously construct n ISTs, it is not fully parallelized for the construction of each spanning tree. In this paper, we present a nonrecursive and fully parallelized approach to construct n ISTs rooted at an arbitrary node of M Q n in O(log N ) time using N nodes of M Q n as processors. In particular, we derive useful properties from the description of paths in ISTs, which make the proof of independency to become easier than ever before.

show abstract

“…In fact, Zehavi and Itai [32] conjectured that for any k-connected graph G and each vertex r of G, there exist k ISTs of G rooted at r. The conjecture has been confirmed only for k-connected graphs with k 6 4 in [3,5,13,32], and it is still open for arbitrary k-connected graphs when k P 5. Moreover, by providing the construction schemes of ISTs, the conjecture has been proved to hold for several restricted classes of graphs or digraphs, such as planar graphs [11,12,20,21], product graphs [1,22], chordal rings [14,28], odd graphs [15], deBruijn and Kautz digraphs [7,10], locally twisted cubes [18], recursive circulant graphs [29,30], and hypercubes [24,31], etc. We refer the readers to a recent paper [25] for additional information regarding recent progress on this problem and current research directions.…”

Section: Introductionmentioning

confidence: 99%