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EDGE NUCLEI FOR SUPERCOMPACT GRAPHS

**Author(s):** Wang Chengde
**Pages:** 1-

7
**Year:** 1988
**Issue:**
4

**Journal:** *Transactions of Beijing Institute of Technology*
**Keyword:** neighborhood; edge nucleus; supercompact; summandable.;
**Abstract:** A graph G is called supercompact if distinct vertices have distinct closed neighborhoods. For a supercompact graph G, an edge e is called removable if G-e is supercompact. The subgraph of G induced by all removable edges is denoted by E0(G ) called the edge nucleus of G. A graph G is called summandable if V(G) can be partitioned into two non-void subsets A and B such that G is the join of G[A] and G[B]. For each integer n>l, we define Ln to be the graph with vertex set V(Ln) = {x1 ,…,xn,b1 ,…, bn, bn+1} and edge set E(Ln) = {xlbi |1≤i≤n} ∪ {b ibj | i≠j}. The results of this paper are the following.Theorem 1. Let G be a connected suparcompact graph with non-void edge nucleus E0. Then |V(G) | ≤2 |V(E0).| - 1,where the equality holds if and only if G is isomorphic to Ln, for some integer n>l.Theorem 2. A graph G is a supercompact summandable graph and E0 is a forest if and only if either G = {x} + P3 or G = (m{x} ∪ nP2) + (m’ {x} ∪ n’P2) where m, n,m’,n’ are non-negative integers with m+n + m’ + n’>2, m + n≠0, m’+n’≠0.

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