  We show that an expected number ratio of radial arcs (gravitationally lensed
images whose major axes lie in the radial direction of a cluster-lens
potential) to tangential arcs (gravitationally lensed images whose major axes
lie in the tangential direction) has strong dependence on the central density
profile of galaxy clusters and has little dependence on other parameters, e.g.
cluster temperature, background source galaxy redshift etc.. A comparison of
the expected number ratios with observed ratios provides a robust test to
constrain the central density profile of galaxy clusters. A tentative
comparison with the observational data shows that the central density profile
of galaxy clusters is r^{-1} - r^{-1.5}. This result indicates that the dark
matter is collisionless at least on cluster scale. Our result gives an upper
limit on a collision cross-section of self-interaction of dark matter
sigma_coll as sigma_coll/m < 0.1 cm^2/g where m is a dark matter particle mass.
