  Let $X$ be an infinite, locally finite, connected, quasi-transitive graph
without loops or multiple edges. A graph height function on $X$ is a map
adapted to the graph structure, assigning to every vertex an integer, called
height. Bridges are self-avoiding walks such that heights of interior vertices
are bounded by the heights of the start- and end-vertex. The number of
self-avoiding walks and the number of bridges of length $n$ starting at a
vertex $o$ of $X$ grow exponentially in $n$ and the bases of these growth rates
are called connective constant and bridge constant, respectively. We show that
for any graph height function $h$ the connective constant of the graph is equal
to the maximum of the two bridge constants given by increasing and decreasing
bridges with respect to $h$. As a concrete example, we apply this result to
calculate the connective constant of the Grandparent graph.
