Geodesic
The Category of Graphs with a Given Subgraph-with Applications to Topology and Algebra
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1506-1517

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By a graph we mean a pair (X, R) where X is a non-void set and R ⊂ X × X. A mapping f: X → Y is called a compatible map (or morphism) from (X, R) into (Y, S) if 2 f(R) ⊂ S, where 2 f: X 2 → Y 2 is defined by 2 f((x1, x2)) = (f(x 1),f(x 2)). The set of all compatible maps from (X, R) into itself forms a monoid (semigroup with a unit element) under composition, which is denoted by M(X, R). A graph (X 1, R 1) is said to be a full subgraph of (X, R) if X 1 ⊂ X and R 1 = R ∩ (X 1 × X 1). A graph (X, R) is said to be without loops if (x, x) ∉ R for all x ∈ X. 
Hedrlín, Z.; Mendelsohn, E. The Category of Graphs with a Given Subgraph-with Applications to Topology and Algebra. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1506-1517. doi: 10.4153/CJM-1969-165-5
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