Possible heights of graph transformation groups

In the following text we prove that for all finite $p\ge 0$ there exists a topological graph $X$ such that $\{p,p+1,p+2,\dots \}\cup \{+\mathrm{\infty }\}$ is the collection of all possible heights for transformation groups with phase space $X$. Moreover for all topological graph $X$ with $p$ as height of transformation group $(Homeo(X),X)$, $\{p,p+1,p+2,\dots \}\cup \{+\mathrm{\infty }\}$ again is the collection of all possible heights for transformation groups with phase space $X$.