Analogous to the fixed point property for ordered sets, a graph has the fixed vertex property if each of its endomorphisms has a fixed vertex. The fixed point theory for ordered sets can be embedded into the fixed vertex theory for graphs. Therefore, the potential for cross-fertilization should be explored.
Order: A Journal on the Theory of Ordered Sets and its Applications
Schröder, B. S.
(2015). The Fixed Vertex Property for Graphs. Order: A Journal on the Theory of Ordered Sets and its Applications, 32(3), 363-377.
Available at: https://aquila.usm.edu/fac_pubs/15690