We present four Mpc-sized radio galaxies which consist of a pair of double-lobed radio sources, aligned along the same axis, and with a coinciding radio core. We have called these peculiar radio sources `double-double radio galaxies (DDRG) and propose a general definition of such sources: A `double-double radio galaxy consists of a pair of double radio sources with a common centre. Furthermore, the two lobes of the inner radio source must have a clearly extended, edge-brightened radio morphology. Adopting this definition we find several other candidate DDRGs in the literature. We find that in all sources the smaller (inner) pair of radio lobes is less luminous than the larger (outer) pair, and that the ratio of 1.4-GHz flux density of these two pairs appears to be anti-correlated with the projected linear size of the inner source. Also, the outer radio structures are large, exceeding 700 kpc. We discuss possible formation scenarios of the DDRGs, and we conclude that an interruption of the jet-forming central activity is the most likely mechanism. For one of our sources (B 1834+620) we have been able to observationally constrain the length of time of the interruption to a few Myr. We discuss several scenarios for the cause of the interruption and suggest multiple encounters between interacting galaxies as a possibility. Finally, we discuss whether such interruptions help the formation of extremely large radio sources.
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