A many-to-many k-disjoint path cover (k-DPC) of a graph G is a set of k disjoint paths joining k sources and k sinks in which each vertex of G is covered by a path. It is called a paired many-to-many disjoint path cover when each source should be joined to a specific sink, and it is called an unpaired many-to-many disjoint path cover when each source can be joined to an arbitrary sink. In this paper, we discuss about paired and unpaired many-to-many disjoint path covers including their relationships, application to strong Hamiltonicity, and necessary conditions. And then, we give a construction scheme for paired many-to-many disjoint path covers in the graph H0oplus H1obtained from connecting two graphs H0and H1with |V(H0)| = |V(H1)| by |V(H1)| pairwise nonadjacent edges joining vertices in H0and vertices in H1, where H0= G0oplus G1and H1= G2oplus G3for some graphs Gj. Using the construction, we show that every m-dimensional restricted HL-graph and recursive circulant G(2m, 4) with f or less faulty elements have a paired k-DPC for any f and k ges 2 with f + 2k les m.

A paired many-to-many k-disjoint path cover (k-DPC) of a graph G is a set of k disjoint paths joining k distinct source-sink pairs in which each vertex of G is covered by a path. This paper is concerned with paired many-to-many disjoint path coverability of hypercube-like interconnection networks, called restricted HL-graphs. The class includes twisted cubes, crossed cubes, multiply twisted cubes, Möbius cubes, Mcubes, and generalized twisted cubes. We show that every restricted HL-graph of degree m with f or less faulty elements has a paired many-to-many k-DPC for any f and k ≥ 2 with f + 2k ≤ m. The result improves the known bound of f + 2k ≤ m − 1 by one.

A many-to-many k-disjoint path cover (k-DPC) of a graph G is a set of k disjoint paths joining k distinct source-sink pairs in which each vertex of G is covered by a path. We deal with the graph G/sub 0/ /spl oplus/ G/sub 1/ obtained from connecting two graphs G/sub 0/ and G/sub 1/ with n vertices each by n pairwise nonadjacent edges joining vertices in G/sub 0/ and vertices in G/sub 1/. Many interconnection networks such as hypercube-like interconnection networks can be represented in the form of G/sub 0/ /spl oplus/ G/sub 1/ connecting two lower dimensional networks G/sub 0/ and G/sub 1/. In the presence of faulty vertices and/or edges, we investigate many-to-many disjoint path coverability of G/sub 0/ /spl oplus/ G/sub 1/ and (G/sub 0/ /spl oplus/ G/sub 1/) /spl oplus/ (G/sub 2/ /spl oplus/ G/sub 3/ ), provided some conditions on the Hamiltonicity and disjoint path coverability of each graph G/sub i/ are satisfied, 0 /spl les/ i /spl les/ 3. We apply our main results to recursive circulant G(2/sup m/, 4) and a subclass of hypercube-like interconnection networks, called restricted HL-graphs. The subclasses includes twisted cubes, crossed cubes, multiply twisted cubes, Mobius cubes, Mcubes, and generalized twisted cubes. We show that all these networks of degree m with f or less faulty elements have a many-to-many k-DPC joining any k distinct source-sink pairs for any k /spl ges/ 1 and f /spl ges/ 0 such that f+2k /spl les/ m - 1.

We study the broadcasting of messages in the single-port, blocking communication model under edge reductions. Edge reductions model the decrease or elimination of broadcast costs on edges. Let T be an n-vertex tree and B be a target broadcast cost. We present an O(n) time algorithm for determining which edges to reduce (i.e., assign cost 0) such that the number of reduced edges is a minimum and a broadcast initiated at a vertex in a determined center set costs at most B. We present an O(n log n) time algorithm to minimize the cost of a broadcast initiated at an arbitrary vertex.

In this paper, we propose and analyze the new interconnection network for parallel computer, called graycube. The graycube has the same number of nodes and edges as hypercube, but it's diameter is about one half of the equivalent hypercube. It has simple recursive structure, routing and broadcasting algorithms. Since hypercube can be embedded into graycube with dialation 2, algorithms developed based on hypercube are easily simulated in graycube. The basic properties, routing and broadcasting algorithms, and hypercube embedding are presented.