Embedding of cycles and wheels into arbitrary trees

Abstract

We estimate and characterize the edge congestion-sum measure for embeddings of various graphs such as cycles, wheels, and generalized wheels into arbitrary trees. All embedding algorithms apply an interesting general technique based on the consecutive label property. Our algorithms produce optimal values of sum of dilations and sum of edge-congestions in linear time.

Bothway embedding of circulant network into grid

“…The wirelength problem [3,4,9,24,28,29] of a graph G into H is to find an embedding of G into H that induces the wirelength WL(G, H). It is interesting to note that the embedding parameters wirelength, dilation sum and congestion sum are all equal [29].…”

“…But the Congestion Lemma and the Partition Lemma [24] have enabled to obtain exact wirelength of embeddings for various architectures [23][24][25]29,[31][32][33]. This technique focuses on specific partitioning of the edge set of the host graph.…”

Wirelength of 1-fault hamiltonian graphs into wheels and fans

“…A wheel graph [26,30] W n of order n is a graph that contains an outer cycle of order n − 1, and for which every vertex in the cycle is connected to one other vertex (which is known as the hub). The edges of a wheel which include the hub are called spokes.…”

Embedding of C_n^2 and C_(n-1)^2+K_1 in to Arbitrary Tree

“…There are also other general results on embeddings [2]. There are algorithms for the embedding of Cycles and wheel into arbitrary tree [30] and k sequential m -ary into hypercube [31]. We apply Lemma 1 for estimation, and use the consecutive label property for characterization.…”