A rational map f:CβC on the Riemann
sphere C is called critically fixed if each critical point
of f is fixed under f. In this article we study properties of a
combinatorial invariant, called Tischler graph, associated with such a map.
More precisely, we show that the Tischler graph of a critically fixed rational
map is always connected, establishing a conjecture made by Kevin Pilgrim. We
also discuss the relevance of this result for classical open problems in
holomorphic dynamics, such as combinatorial classification problem and global
curve attractor problem