On identifying codes in the king grid that are robust against edge deletions Assume that G  (V, E) is an undirected graph, and C subseteq V. For every v in V, we denote I_r(G;v)   u in C: d(u,v) leq r, where d(u,v) denotes the number of edges on any shortest path from u to v. If all the sets I_r(G;v) for v in V are pairwise different, and none of them is the empty set, the code C is called ridentifying. If C is ridentifying in all graphs Gu0027 that can be obtained from G by deleting at most t edges, we say that C is robust against t known edge deletions. Codes that are robust against t unknown edge deletions form a related class. We study these two classes of codes in the king grid with the vertex set Bbb Z2 where two different vertices are adjacent if their Euclidean distance is at most sqrt2.

Hitting minors on bounded treewidth graphs. II. Singleexponential algorithms Abstract   For a finite collection of graphs    F   , the    F    MDeletion  (resp.    F    TMDeletion ) problem consists in, given a graph G and an integer k, decide whether there exists    S  xe2x8ax86  V  (  G  )    with      S    xe2x89xa4  k    such that    G  xe2x88x96  S    does not contain any of the graphs in    F    as a minor (resp. topological minor). We are interested in the parameterized complexity of both problems when the parameter is the treewidth of G, denoted by    tw   , and specifically in the cases where    F    contains a single connected planar graph H. We present algorithms running in time      2    O  (   tw   )    xe2x8bx85    n    O  (  1  )     , called singleexponential, when H is either      P    3     ,      P    4     ,      C    4     , the  paw , the  chair , and the  banner  for both      H      MDeletion  and      H      TMDeletion , and when    H      K    1  ,  i     , with    i  xe2x89xa5  1   , for      H      TMDeletion . Some of these algorithms use the rankbased approach introduced by Bodlaender et al. (2015)  7 . This is the second of a series of articles on this topic, and the results given here together with other ones allow us, in particular, to provide a tight dichotomy on the complexity of      H      MDeletion  in terms of H.