Abstract:

      Let G be a finite, simple and undirected graph and U be its edge subset If G - U is disconnected and each component of G - U contains at least four vertices, then such an edge set is 4-restricted edgecut of G. The 4-restricted edgecut U whose cardinality is the smallest is called a λ₄- cut, and its cardinality is called the 4-restricted edgeconnectivity, denoted by λ₄=λ₄(G) . G is  λ₄- optimal if λ₄=ξ₄(G)  and super -λ₄  if every λ₄-cut isolates a connected subgraph of order 4. This paper presents some sufficient conditions for a λ₄- optimal and super -λ₄ graph with the neighborhood intersection condition.

Key words: graph, 4-restricted edge connectivity, λ₄- optimal graph, super -λ₄ graph, neighborhood