Subgraph fault tolerance of distance optimally edge connected hypercubes and folded hypercubes Abstract   Hypercube and folded hypercube are the most fundamental interconnection networks for the attractive topological properties. We assume for any distinct vertices     u  ,  v  xe2x88x88  V  ,  xcexba   (  u  ,  v  )      defined as local connectivity of    u    and    v   , is the maximum number of independent     (  u  ,  v  )    paths in    G   . Similarly,     xcexbb   (  u  ,  v  )      is local edge connectivity of     u  ,  v    . For some     t  xe2x88x88     1  ,  D   (  G  )      ,    xe2x88x80    u  ,  v  xe2x88x88  V  ,  u  xe2x89xa0  v    , and     d   (  u  ,  v  )     t    , if     xcexba   (  u  ,  v  )    (  o  r  xcexbb   (  u  ,  v  )   )     m  i  n     d   (  u  )   ,    d   (  v  )        , then    G    is    t   distance optimally (edge) connected, where     D   (  G  )      is the diameter of    G    and     d   (  u  )      is the degree of    u   . For all integers     0    k  xe2x89xa4  t    , if    G    is    k   distance optimally connected, then we call    G    is    t   distance local optimally connected. Similarly, we have the definition of    t   distance local optimally edge connected.  In this paper, we show that after deleting       Q    k     (  k  xe2x89xa4  n  xe2x88x92  1  )     ,       Q    n    xe2x88x92    Q    k       and     F    Q    n    xe2x88x92    Q    k       are 2distance local optimally edge connected.

Small Cuts and Connectivity Certificates: A Fault Tolerant Approach We revisit classical connectivity problems in the CONGEST model of distributed computing. By using techniques from fault tolerant network design, we show improved constructions, some of which are even local (i.e., with widetildeO(1) rounds) for problems that are closely related to hard global problems (i.e., with a lower bound of Omega(Diamsqrtn) rounds).  :52,main results are:  :56,For Ddiameter unweighted graphs with constant edge connectivity, we show an exact distributed deterministic computation of the minimum cut in poly(D) rounds. This resolves one the open problems recently raised in Daga, Henzinger, Nanongkai and Saranurak, STOCu002719.  :56,For Ddiameter unweighted :94,graphs, we present a deterministic algorithm that computes of all edge connectivities up to constant in poly(D)cdot 2O(sqrtlog nloglog n) rounds.  :119,Computation of sparse lambda connectivity certificates in widetildeO(lambda) rounds. Previous constructions where known only for lambda leq 3 and required O(D) rounds. This resolves the problem raised by Dori PODCu002718.