We study the "combinatorial anabelian geometry" that governs the relationship between the dual semi-graph of a pointed stable curve and various associated profinite fundamental groups of the pointed stable curve. Although many results of this type have been obtained previously in various particular situations of interest under unnecessarily strong hypotheses, the goal of the present paper is to step back from such "typical situations of interest" and instead to consider this topic in the abstract -- a point of view which allows one to prove results of this type in much greater generality under very weak hypotheses.