Joint statement condemns Edinburgh Derby
The Edinburgh Derby, which ended 0-0, was dull of drama as Police Scotland made five arrests.
Hibernian manager Neil Lennon was hit by a coin during the match, and Hearts goalkeeper Zdenek Zlamal appeared to be struck by a fan when he attempted to retrieve the ball.
The clubs released a joint statement condemning the actions: "The two clubs are united in their condemnation of the isolated events which marred an otherwise compelling fixture, and will together work with Police Scotland to ensure those individuals concerned are identified and face appropriate sanctions."
Dempster said: "The two clubs are united. This kind of behaviour is simply not acceptable. We do everything we can to ensure that we run matches in a way that is safe for everyone involved. Players, coaches, supporters, the media, everyone who goes to a football match is entitled to enjoy the match in safety and security.
"We will not allow the mindless actions of a few foolish individuals to jeopardise that. We will learn any lessons that we can from this, but we would also call on supporters to play their part in helping us to identify those responsible for this kind of unacceptable and potentially dangerous behaviour.
"The two Clubs work together in all kinds of ways, and there is great mutual respect. There is an intense sporting rivalry, and we all enjoy that and would not wish to dilute it, but it needs to be a respectful rivalry."