  Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and
Seymour [Approximating clique-width and branch-width. J. Combin. Theory Ser. B,
96(4):514--528, 2006]. Motivated from recent development on graph modification
problems regarding classes of graphs of bounded treewidth or pathwidth, we
study the Linear Rankwidth-1 Vertex Deletion problem (shortly, LRW1-Vertex
Deletion). In the LRW1-Vertex Deletion problem, given an $n$-vertex graph $G$
and a positive integer $k$, we want to decide whether there is a set of at most
$k$ vertices whose removal turns $G$ into a graph of linear rankwidth at most
$1$ and find such a vertex set if one exists. While the meta-theorem of
Courcelle, Makowsky, and Rotics implies that LRW1-Vertex Deletion can be solved
in time $f(k)\cdot n^3$ for some function $f$, it is not clear whether this
problem allows a running time with a modest exponential function.
  We first establish that LRW1-Vertex Deletion can be solved in time $8^k\cdot
n^{\mathcal{O}(1)}$. The major obstacle to this end is how to handle a long
induced cycle as an obstruction. To fix this issue, we define necklace graphs
and investigate their structural properties. Later, we reduce the polynomial
factor by refining the trivial branching step based on a cliquewidth expression
of a graph, and obtain an algorithm that runs in time $2^{\mathcal{O}(k)}\cdot
n^4$. We also prove that the running time cannot be improved to $2^{o(k)}\cdot
n^{\mathcal{O}(1)}$ under the Exponential Time Hypothesis assumption. Lastly,
we show that the LRW1-Vertex Deletion problem admits a polynomial kernel.
