The total bondage number of grid graphs

The total domination number of a graph GG without isolated vertices is the minimum number of vertices that dominate all vertices in GG. The total bondage number bt(G)bt(G) of GG is the minimum number of edges whose removal enlarges the total domination number. This paper considers grid graphs. An (n,m)(n,m)-grid graph Gn,mGn,m is defined as the cartesian product of two paths PnPn and PmPm. This paper determines the exact values of bt(Gn,2)bt(Gn,2) and bt(Gn,3)bt(Gn,3), and establishes some upper bounds of bt(Gn,4)bt(Gn,4).