The (1,2)-Step Competition Graph of a Round Out-Tournament

Originally derived from the idea of food webs of predator and prey, the relationship between digraphs and their competition graph has been a wonder in the world of mathematics since 1962. An exploration of round digraphs, competition graphs and (1,2)-step competition graphs. We show that round digraphs are out-tournaments, and find patterns between the competition graphs and (1,2)-step competition graphs of out-tournaments. Round digraphs have the property that the arcs from each vertex follow a clockwise pattern. A tournament is a digraph with the property that there is a single arc between any two vertices. An out-tournament is a digraph where the outset of each vertex is a tournament. From these unique direction graphs, we obtain the (1,2)- step competition graph; a graph that not only has edges between two competing vertices, but also between two vertices that compete through another vertex. We observe the patterns found between the different number of arcs of each vertices and the number of edges in the corresponding (1,2)-step competition graph