We prove that cycles, wheels and block graphs have sequentially Cohen-Macaulay binomial edge ideals. Moreover, we provide a construction of new families of sequentially Cohen-Macaulay graphs by cones.
Comment: In this version we have made a substantial correction of the Introduction. We also improved the overall notation. Moreover, to improve the readability and comprehension, we added Example 2.4, 3.2, 4.3, 4.4 and Remark 4.2