Sentence:  First, let us define its 1-dimensional analog, that is, a topological graph. A <WSD>"graph</WSD>" \Delta is a 1-dimensional stratified topological space with finitely many 0-strata (vertices) and finitely many 1-strata (edges)....A <WSD>graph</WSD> such that any vertex belongs to at least two half-edges we call an "s-<WSD>graph</WSD>". Clearly the boundary \partial\Omega of a surface \Omega with marked points is an s-<WSD>graph</WSD>. / A morphism of <WSD>graphs</WSD> \varphi: \Delta'\rightarrow\Delta" is a continuous epimorphic map of <WSD>graphs</WSD> compatible with the stratification; i.e., the restriction of \varphi to any open 1-stratum (interior of an edge) of \Delta' is a local (therefore, global) homeomorphism with appropriate open 1-stratum of \Delta". 
Meaning of the word marked with <WSD> tags:  A topological space which represents some graph (ordered pair of sets) and which is constructed by representing the "vertices" as "points" and the "edges" as copies of the real interval [0,1] (where, for any given edge, 0 and 1 are identified with the points representing the two vertices) and equipping the result with a particular topology called the "graph topology".