Restricted arc-connectivity of unidirectional star graphs

Abstract

The restricted arc-connectivity is an effective assessment of the reliability of directed networks, which is an extended notion of arc-connectivity. Let D be a digraph. An arc set S of D is a restricted arc-cut of D if D−S has a strong connected component D′ such that |V(D′)|≥2 and D−V(D′) contains an arc. The digraph D is called λ′-connected if there exists a restricted arc-cut in D. The restricted arc-connectivity λ′(D) of a λ′-connected digraph D is the minimum cardinality over all restricted arc-cuts. We can get a unidirectional star graph by orienting the star graph with Day-Tripathi orientation. In this paper, we first show that the restricted arc-connectivity of the n-dimensional unidirectional star graph is n−2 when n is odd and n−3 when n is even for n≥4. As a consequence, we prove that n-dimensional unidirectional star graph is super-λ when n≥3 and n≠4.