the following definition is an edge-centric definition of graphs, in which edges are primary and vertices are defined in terms of edges.
for any finite set , a graph on is a pair where is a partition of the set , called the dart set of . that is, is a collection of disjoint, nonempty, mutually exhaustive subsets of . each subset is a vertex of . for any , the darts of are the pairs and , of which the primary dart of is . for brevity, we can write as and as .
the head of a dart is the block such that contains . the tail of is the head of .
there is one seeming disadvantage: this definition of graphs does not permit the existence of isolated vertices, vertices with no incident edges. this disadvantage is mitigated by interpreting a subset of edges of a graph as a kind of subgraph.
an embedded graph with its vertices listed:
define the bijection rev on darts by . for a dart , is called the reverse of .
the terminology dart reverse may be deceiving, consider the edges , their corresponding darts are , we have that but because .