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Independent Spanning Trees With Small Depths in Iterated Line Digraphs
, 1997
"... . We show that the independent spanning tree conjecture on digraphs is true if we restrict ourselves to line digraphs. Also, we construct independent spanning trees with small depths in iterated line digraphs. From the results, we can obtain independent spanning trees with small depths in de Bruijn ..."
Abstract

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. We show that the independent spanning tree conjecture on digraphs is true if we restrict ourselves to line digraphs. Also, we construct independent spanning trees with small depths in iterated line digraphs. From the results, we can obtain independent spanning trees with small depths in de Bruijn and Kautz digraphs that improve the previously known upper bounds on the depths. Keywords. independent spanning trees, line digraphs, vertexconnectivity, de Bruijn digraphs, Kautz digraphs, interconnection networks, broadcasting. 1 Introduction Unless stated otherwise each digraph of this paper is nite and may have loops but not multiarcs. Let G be a digraph. Then V (G) and A(G) denote the vertex set and the arc set of G, respectively. Let (u; v) 2 A(G). Then we say that u is adjacent to v, and v is adjacent from u. Also, it is said that (u; v) is incident to v and incident from u. Let (v; w) 2 A(G). Then we say that (u; v) is adjacent to (v; w), and (v; w) is adjacent from (u; v). Let ...
The 24th Workshop on Combinatorial Mathematics and Computation Theory Parallel Construction of Independent Spanning Trees on Multidimensional Tori ∗
"... A set of spanning trees rooted at vertex r in G is called independent spanning trees (IST) if for each vertex v in G, v = r, the paths from v to r in any two trees are different and vertexdisjoint. If the connectivity of G is k, the IST problem is to construct k IST rooted at each vertex. The IST p ..."
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A set of spanning trees rooted at vertex r in G is called independent spanning trees (IST) if for each vertex v in G, v = r, the paths from v to r in any two trees are different and vertexdisjoint. If the connectivity of G is k, the IST problem is to construct k IST rooted at each vertex. The IST problem has found applications in faulttolerant broadcasting, but it is still open for general graph with connectivity greater than four. Obokata et al. [IEICE Trans. Fundamentals of Electronics, Communications and Computer Sciences E79A (1996) 1894–1903] have proved that the IST problem can be solved on multidimensional tori. However, their construction algorithm forbids the possibility of parallel processing. In this paper, we shall propose a parallel algorithm that is based on the Latin square scheme to solve the IST problem on multidimensional tori.
The 25th Workshop on Combinatorial Mathematics and Computation Theory On the Independent Spanning Trees of Recursive Circulant Graphs G(cd m, d) with d � 3 ∗
"... Two spanning trees of a graph G are said to be independent if they are rooted at the same vertex r, and for each vertex v = r in G, the two different paths from v to r, one path in each tree, are internally disjoint. A set of spanning trees of G is independent if they are pairwise independent. A re ..."
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Two spanning trees of a graph G are said to be independent if they are rooted at the same vertex r, and for each vertex v = r in G, the two different paths from v to r, one path in each tree, are internally disjoint. A set of spanning trees of G is independent if they are pairwise independent. A recursive circulant graph G(N, d) has N = cdm vertices labeled from 0 to N − 1, where d � 2, m � 1, and 1 � c < d, and two vertices x, y ∈ G(N, d) are adjacent if and only if there is an integer k with 0 � k � ⌈logd N⌉−1 such that x±dk ≡ y (mod N). In this paper, we propose an algorithm to construct multiple independent spanning trees on recursive circulant graphs G(cdm, d) under the condition d � 3, where the number of independent spanning trees matches the connectivity of G(cdm, d).