On the split reliability of graphs

A common model of robustness a graph against random failures has all vertices operational, but the edges independently operational with probability p $$ . One can ask for that communicate (all-terminal reliability) or two specific (or terminals) each other (two-terminal reliability). relatively new measure is split reliability, where fixed s and t , we consider every vertex communicates one not both. In this article, explore existence numbers n ≥ 2 n\ge m − 1 m\ge n-1 an optimal connected ( ) \left(n,m\right) -graph G {G}_{n,m} is, which any such H reliability at least as large values ∈ [ 0 ] p\in \left[0,1\right] Unlike similar problems all-terminal two-terminal only partial results are known, completely solve issue show there if ≤ 3 n\le = m=n-1 4 n=m=4