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The superior complement in graphs
Authors: KM. Kathiresan, G. Marimuthu, C. Parameswaran
Number of views: 366
For distinct vertices u and v of a nontrivial connected graph G, we let Du,v = N[u] [ N[v]. We define a Du,v-walk as a u-v walk in G that contains every vertex of Du,v. The superior distance dD(u, v) from u to v
is the length of a shortest Du,v-walk. For each vertex u 2 V (G), define d−D(u) = min{dD(u, v) : v 2 V (G) − {u}}. A vertex v(6= u) is called a superior neighbor of u if dD(u, v) = d−D(u). In this paper we define the concept
of superior complement of a graph G as follows: The superior complement of a graph G is denoted by GD whose vertex set is as in G. For a vertex u, let Au = {v 2 V (G) : dD(u, v) d−D(u) + 1}. Then u is adjacent to all the
vertices v 2 Au in GD. The main focus of this paper is to prove that there is no relationship between the superior diameter dD(G) of a graph G and the superior diameter dD(GD) of the superior complement GD of G.