Graffiti is more generally known as criminal damage. Damaging someones property can be a misdemeanor or a felony, depending on the dollar amount of the damage. (A.R.S. § 13-1602).

A charge of criminal damage can be made against you when the prosecution believes it has enough evidence to have a judge or jury convinced beyond a reasonable doubt that you committed the offense. The officer who arrests you does not have to witness the damage when you did it. In fact, there doesnt have to be an eye-witness to the offense at all.

It is possible, for example, that someone else saw you commit the crime and reported it to the police. It is also conceivable that the officer saw the graffiti on a wall, and when questioning you about something else, noticed that your notebook had similar drawings and sayings as the graffiti. The state would still have to be able to convince the judge or jury that you did it, however, for you to be found guilty. The charge, by itself, is not evidence.