A Linear Kernel for Planar Semitotal Domination

A set of vertices $S$ in a graph $G$ is a semitotal dominating set if $S$ is a dominating set and every vertex in $S$ has a witness, that is, another vertex in $S$ within distance at most two. The Semitotal Domination problem asks whether $G$ admits such a set of size at most $k$. We study this problem parameterized by the solution size. We provide a linear kernel of size $358k$ on planar graphs, a result that adds to several existing linear kernels for other domination variants on planar graphs. Additionally, we show Semitotal Domination is $W[2]$-hard on split graphs, on bipartite graphs and on $t$-claw-free graphs, it is $W[1]$-hard on circle graphs.