Abstract
This paper presents some graph-theoretic questions from the viewpoint of the portion of category theory which has become common knowledge. In particular, the reader is encouraged to consider whether there is only one natural category of graphs and how theories of directed graphs and undirected graphs are related.
Highlights
This paper presents some graph-theoretic questions from the viewioint of the portion of category theory which has become common knowledge
We shall present evidence that, as originally presented, these definitions were wrong since they did not account for the category theoretical properties which have been recognized as essential parts of the algebraic structure and cartesian closedness
The difficulty with a study of a "category of graphs" is that graph theory was born as a branch of
Summary
FI- N [A__P,Ens] maps each At-graph G to the quotient A-graph G1 with the same vertices as G, the same edges between distinct vertices, but having only one loop at each vertex at which G has loops and having no new loops. The coreflective subcategory of F I consisting of A-graphs with exactly one loop per vertex is equivalent to the reflective subcategory A of B-graphs with only degenerate loops. 2 [A_P,Ens] A second important subcategory of is the full reflective subcategory generated by the simple A_-graphs G with at most one edge from any particular vertex to another. Using an argument similar to the one for it follows that there exists a reflective subcategory A2 of [Bp,Ens] which is 2 equivalent to a coreflective subcategory of [_A_P,Ens] Theorem 2.1 establishes equivalences between subcategorles of and [BP,Ens] which introduce the combinatorially useful concepts of "absence of (unnecessary) loops" or "characterization of edges by their origin and terminal vertices". Either of the two possible general definitions of "directed graph" leads to the same theory. Though, we have found that the interplay of these generalizations leads to deeper properties than one might have expected
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