  In this paper, we study the game of cops and robber on the class of graphs
with no even hole (induced cycle of even length) and claw (a star with three
leaves). The cop number of a graph $G$ is defined as the minimum number of cops
needed to capture the robber. Here, we prove that the cop number of all
claw-free even-hole-free graphs is at most two and, in addition, the capture
time is at most $2n$ rounds, where $n$ is the number of vertices of the graph.
Moreover, our results can be viewed as a first step towards studying the
structure of claw-free even-hole-free graphs.
