Abstract
In this paper we consider the monophonic embedding ofcirculant networks into cycles and we produce an algorithmto get the monophonic wirelength of the same.Further wend that the monophonic wirelength of some family of cir-In this paper, we define the monophonic embedding of graph G into another graph H and this paper presents a monophonic algorithm to find the monophonic wirelength of circulant networks G[n, ±S], where S ⊆ {1,2,3,…,n/2} into the family of Cycle Cn, n≥ 4. The mono-phonic embedding of a graph G into a graph H is an embedding denoted by fmis a bijective map from the vertex set of G into the vertex set of H and fm is a one-one mapping from the edge set (x, y) of G into Pm(H) where Pm(H) is the set of monophonic paths between fm(x) and fm(y) for every fm(x), fm(y) ∈ H. The monophonic wirelength of fm of G into H is the sum of distances of monophonic paths between two vertices fm(x) and fm(y) in H such that (x, y) ∈ E(G). In addition, the eccentricity, radius and diameter of an embedding of G into H are defined. The average wirelength of an embedding is defined and the bounds of average wirelength of some embeddings have been found.
Highlights
For vertices u and v in a connected graph G, The distance d(u, v) is the length of the shortest u-v path in G
We define the monophonic embedding of graph G into another graph H and this paper presents a monophonic algorithm to find the monophonic wirelength of circulant networks G[n, ±S], where S ⊆ {1,2,3,...,n/2} into the family of Cycle Cn, n≥ 4
The mono-phonic embedding of a graph G into a graph H is an embedding denoted by fmis a bijective map from the vertex set of G into the vertex set of H and fm is a one-one mapping from the edge set (x, y) of G into Pm(H) where Pm(H) is the set of monophonic paths between fm(x) and fm(y) for every fm(x), fm(y) ∈ H
Summary
For vertices u and v in a connected graph G, The distance d(u, v) is the length of the shortest u-v path in G. For two vertices u and v in a connected graph G, the monophonic distance dm(u, v) is the length of the longest u - v monophonic path in G. An embedding f of G into H is defined as follows: 1) f is a bijective map from V(G) to V(H). An embedding fm: G → H is called a monophonic embedding if fm maps each vertex of G into a vertex of H and each edge (x, y) of G is mapped to a monophonic path between fm(x) and fm(y) in H. The wirelength problem of a graph G into H is to find an embedding of G into H that induces the minimum wirelength WL(G, H) as defined in [6],[11], [12]
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