The t/k-diagnosability and strong Menger connectivity on star graphs with conditional faults

Abstract

The t/k-diagnosis, a famous diagnosis strategy, is proposed by Somani and Peleg. In this strategy, all the faulty vertices, no more than t, can be isolated into a faulty set, which may contain no more than k fault-free vertices. Somani and Peleg ([16], 1996) first proved that Sn was [(k+1)n−3k−2]/k-diagnosable for 0<k≤n. Chen and Liu ([4], 2012) found that the proof of that proposition was flawed and they pointed out that Sn was actually [(k+1)n−3k−1]/k-diagnosable for small k without a specific rang of k. Zhou et al. ([21], 2015) proved that Sn(n≥5) was [(k+1)n−3k−1]/k-diagnosable for 1≤k≤3 and they proposed a problem: was Sn still [(k+1)n−3k−1]/k-diagnosable for k≥4? In this paper, we prove that Sn(n≥5) is (5n−14)/4-diagnosable and it is not (5n−13)/4-diagnosable, which gives a negative answer to the problem of Zhou et al. Furthermore, we show that Sn is still strongly Menger connected even if (4n−13) vertices fail, which means there are min⁡{degG−F⁡(x),degG−F⁡(y)}-internally vertex disjoint paths between any two different vertices x,y in G−F for F∈V(Sn), δ(G−F)≥2 and |F|≤4n−13, and the bound 4n−13 is sharp.