Constructing Three Completely Independent Spanning Trees in Locally Twisted Cubes

• Published in: Frontiers in Algorithmics Vol. 11458; pp. 88 - 99
• Main Authors: Pai, Kung-Jui, Chang, Ruay-Shiung, Chang, Jou-Ming, Wu, Ro-Yu
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2019; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Completely independent spanning trees; Diameter; Interconnection networks; Locally twisted cubes
• ISBN: 3030181251; 9783030181253
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-030-18126-0_8

For the underlying graph G of a network, k spanning trees of G are called completely independent spanning trees (CISTs for short) if they are mutually inner-node-disjoint. It has been known that determining the existence of k CISTs in a graph is an NP-hard problem, even for k=2 $$k=2$$ . Accordingly, researches focused on the problem of constructing multiple CISTs in some famous networks. Pai and Chang [28] proposed a unified approach to recursively construct two CISTs with diameter 2n-1 $$2n-1$$ in several n-dimensional hypercube-variant networks for n⩾4 $$n\geqslant 4$$ , including locally twisted cubes LTQn $$LTQ_n$$ . Later on, they provided a new construction for LTQn $$LTQ_n$$ and showed that the diameter of two CISTs can be reduced to 2n-2 $$2n-2$$ if n=4 $$n=4$$ (and thus is optimal) and 2n-3 $$2n-3$$ if n⩾5 $$n\geqslant 5$$ . In this paper, we intend to construct more CISTs of LTQn $$LTQ_n$$ . We develop a novel tree searching algorithm, called two-stages tree-searching algorithm, to construct three CISTs of LTQ6 $$LTQ_6$$ and show that the three CISTs of the high-dimensional LTQn $$LTQ_n$$ for n⩾7 $$n\geqslant 7$$ can be constructed by recursion. The diameters of three CISTs for LTQn $$LTQ_n$$ we constructed are 9, 12 and 14 when n=6 $$n=6$$ , and are 2n-3 $$2n-3$$ , 2n-1 $$2n-1$$ and 2n+1 $$2n+1$$ when n⩾7 $$n\geqslant 7$$ .