Segregated extension of graphs

Abstract

A graph in which any two adjacent vertices have distinct degrees is totally segregated. In this article segregating sequence, which is a new tool for finding segregated extension of given graph is introduced. If G is an undirected graph which contains a vertex v, then the graph G◦v is obtained from G by adding a new vertex v’ which is connected to all the neighbors of v. More generally, if v 1 , v 2 ,··· , v n are the vertices of G and t = (t 1 ,t 2 ,··· ,t n ) is a vector of positive integers then H = G◦t is constructed by substituting for each v i an independent set of t i vertices v 1 i , v 2 i ,··· , v ti i and joining v s i with v t j if and only if v i and v j are adjacent in G. If G is not totally segregated and G◦t is totally segregated, then the sequence t is a segregating sequence of G. Here it is proved that any graph can be embedded as an induced subgraph in a totally segregated graph. Further, segregating sequence for many classes of graphs are determined.