Let $X$ be a graph and let $\CC$ be a class of $X$-maps (that is, of continuous maps from $X$ into itself). A map $f\in \CC$ is said to have full periodicity if $\Per(f)=\N$ (here, $\Per(f)$ denotes the set of periods of all periodic points of $f$ and $\N$ the set of positive integers). The set $K\subseteq \N$ is a full periodicity kernel of $\CC$ if it satisfies the following two conditions: (i) If $f\in \CC$ and $K\subseteq \Per(f)$ then $f$ has full periodicity and (ii) if $S\subset \N$ is a set such that, for every $f\in \CC$, $S\subseteq \Per(f)$ implies $\Per(f)=\N$, then $K\subseteq S$. In this paper we show the existence and characterize the full periodicity kernel of the class of continuous maps from a graph with zero Euler characteristic to itself having all branching points fixed.