Abstract
A set of the spanning trees in a graph <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is called independent spanning trees if they have a common root <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> and for each vertex <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v\in V(G)\setminus \{r\}$ </tex-math></inline-formula> , the paths from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> in any two trees are directed edge-disjoint and internally vertex-disjoint. The construction of independent spanning trees has many practical applications in reliable communication networks, such as fault-tolerant transmission and secure message distribution. A burnt pancake network <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$BP_{n}$ </tex-math></inline-formula> is a kind of Cayley graph, which has been proposed as the topology of an interconnection network. In this paper, we provide a two stages construction scheme that can be used to construct a maximal number of independent spanning trees on a burnt pancake network in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(N\times n)$ </tex-math></inline-formula> time, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> is the number of nodes of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$BP_{n}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is the dimension of the network. Furthermore, we prove the correctness of our proposed algorithm in constructing independent spanning trees.
Highlights
The design of modern interconnected networks faces several critical demands, such as how to perform fault-tolerant transmission and secure message distribution in a reliable communication network
Disjoint paths could be used in secure message distribution over a fault-free network in the following way [1], [33]
Zehavi and Itai [42] conjectured that there exist k ISTs rooted at an arbitrary vertex in a k-connected graph. This conjecture has been confirmed only for k-connected graphs with k ≤ 4. Since this conjecture is still unsolved for the general k-connected graphs of k ≥ 5, the follow-up researchers mainly focused on the study of constructing multiple ISTs on specific interconnection networks
Summary
The design of modern interconnected networks faces several critical demands, such as how to perform fault-tolerant transmission and secure message distribution in a reliable communication network. This conjecture has been confirmed only for k-connected graphs with k ≤ 4 (see [9], [11], [19]) Since this conjecture is still unsolved for the general k-connected graphs of k ≥ 5, the follow-up researchers mainly focused on the study of constructing multiple ISTs on specific interconnection networks. Note that there is a similar problem called the construction of completely independent spanning trees (CISTs for short) in a network. Let BPn denote the n-dimensional burnt pancake graph with the following definition: (i) the addresses of all nodes are permutations of n elements from 1 to n, where n ∈ N; (ii) each element is signed or unsigned; and (iii) every node is adjacent to some other node such that addresses between them are taken prefix reversal and sign reversal [14].
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