Restricted edge connectivity of graphs on degree

• Published in: Journal of intelligent & fuzzy systems 2018-01, Vol.35 (4), p.3955-3958
• Main Authors: Guo, Litao, Lin, Bernard L.S.
• Format: Article
• Language: English
• Subjects: Graph theory; Graphs
• ISSN: 1064-1246; 1875-8967
• DOI: 10.3233/JIFS-169718

Let G = (V, E) be a connected graph. An edge set S ⊂ E is a 3-restricted edge cut, if G - S is disconnected and every component of G - S has at least three vertices. The 3-restricted edge connectivity λ3 (G) of G is the cardinality of a minimum restricted edge cut of G. A graph G is λ3-connected, if 3-restricted edge cuts exist. A graph G is called λ′-optimal, if λ3 (G) = ξ3 (G), where ξ 3 ( G ) = min { | [ X , X ¯ ] | : X ⊂ V , | X | = 3 , G [ X ] is connected}. Furthermore, if every minimum 3-restricted edge cut is a set of edges incident to a set of three vertices, then G is said to be super 3-restricted edge connected or super-λ3 for simplicity. Inverse degree of G is R ( G ) = ∑ v ∈ V 1 d ( v ) , where d (v) denotes the degree of the vertex v. In this paper, we study the relation of inverse degree and super 3-restricted edge connected graphs.