As Clanad says, suckers come out from below the graft. Usually they have 7 leaves on each leaf-stem and the foliage is lighter in colour than the main plant.
I would disagree with Clanad on replacing the rose rather than removing the suckers (if that's what they are) - replacing roses can be tricky because of "rose-sickness". Planting a new rose in place of an old one very often results in the new rose not becoming a good plant. We have around 100 roses round our garden which we planted when we moved in 35 years ago. We tried replacing a few which were getting weary and the replacments nver formed good plants. We've tried again this year replacing 4 and trying some of this stuff:
http://rootgrow.co.uk/
We only planted the replacements at the beginning of the year but all four of them seem to be doing well.