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Optimal orientations of graphs with edges added
Author
Lee, Hwee Yan
Supervisor
Tay, Eng Guan
Abstract
For a graphy G, let D(G) be the family of strong orientations of G, and define the orientation number: d(G) = min {d(D)
D ε D(G)}, where d(D) is the diameter of the digraph D. We shall call any orientation D in D(G) with d(D) = d(G) an optimal orientation of G. In this academic exercise, we consider two graphs, namely complete bipartite graph and cycle graph to find out their optimal orientations with edges added. We shall first define the following: Let G = G1 + G2 denote the addition of two graphs G1 and G2 such that V(G) = V(G1) ◡ V(G2) and E(G) = E(G1) ◡ E(G2) ◡ uv for all u ε E(G1) and v ε E(G2). Let G=H+e denote the graph H such that V(G)=V(H) and E(G)=E(H) ◡ e.
We have only considered the complete bipartite graphs with edges added to the partite set of order m to obtain a new graph Km + Nn. We can reduce the orientation number from either four or three to two, but we are particularly interested in the upper bounds of n for a fixed m given that the graph Km + Nn admits an orientation of diameter two. For the cycle Cn, we want to find the optimal orientations of Cn, where n ≥ 4, with at most two added edges, that is, we only consider the families of graphs Cn + e and Cn + e1 + e2. We want to investigate if these families of graphs will yield smaller orientation numbers than Cn.
We have only considered the complete bipartite graphs with edges added to the partite set of order m to obtain a new graph Km + Nn. We can reduce the orientation number from either four or three to two, but we are particularly interested in the upper bounds of n for a fixed m given that the graph Km + Nn admits an orientation of diameter two. For the cycle Cn, we want to find the optimal orientations of Cn, where n ≥ 4, with at most two added edges, that is, we only consider the families of graphs Cn + e and Cn + e1 + e2. We want to investigate if these families of graphs will yield smaller orientation numbers than Cn.
Date Issued
2002
Call Number
QA166 Lee
Date Submitted
2002