## Poster Session #1: UC Ballroom

Poster

Kim Factor

#### Faculty Mentor’s Department

Dept. of Mathematics, Statistics, and Computer Science

#### Abstract / Artist's Statement

Graph theory is a useful tool for studying systems of food webs, a concept from ecology that models the predator-prey relationships between species in an ecosystem. We have used this concept to inform and motivate our exploration of graph theory. In particular we examine the characteristics of (1,2)-step competition graphs developed by Factor and Merz in 2010, which are an extension of normal competition graphs first introduced by Cohen in 1968. Factor and Merz define the (1,2)-step competition graph of a digraph D, denoted by C1,2(D), as the graph with the same vertex set as D and an edge between vertices x and y if and only if there exists some z in V(D) for which either dD\{x}(y, z) = 1 and dD\{y}(x, z) =< 2 or, dD\{y}(x, z) = 1 and dD\{x}(y, z) =< 2. We extend this definition and say that given x, y in V(D) such that (x, y) in E(C1,2), x and y compete directly if there exists a vertex z in D such that d(y, z) = 1 and d(x, z) = 1. We then call the edge (x, y) in E(C1,2) a direct competition between x and y. Otherwise, we say that x and y compete indirectly and we call the edge (x, y) in E(C1,2) an indirect competition between x and y. We have developed a family of digraphs that induce complete components in their (1,2)-step competition graphs that appear to have a minimum number of direct competitions.

#### Category

Physical Sciences

#### Share

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Apr 11th, 11:00 AM Apr 11th, 12:00 PM

Minimizing Direct Competitions in Complete Components of (1,2)-Step Competition Graphs

Graph theory is a useful tool for studying systems of food webs, a concept from ecology that models the predator-prey relationships between species in an ecosystem. We have used this concept to inform and motivate our exploration of graph theory. In particular we examine the characteristics of (1,2)-step competition graphs developed by Factor and Merz in 2010, which are an extension of normal competition graphs first introduced by Cohen in 1968. Factor and Merz define the (1,2)-step competition graph of a digraph D, denoted by C1,2(D), as the graph with the same vertex set as D and an edge between vertices x and y if and only if there exists some z in V(D) for which either dD\{x}(y, z) = 1 and dD\{y}(x, z) =< 2 or, dD\{y}(x, z) = 1 and dD\{x}(y, z) =< 2. We extend this definition and say that given x, y in V(D) such that (x, y) in E(C1,2), x and y compete directly if there exists a vertex z in D such that d(y, z) = 1 and d(x, z) = 1. We then call the edge (x, y) in E(C1,2) a direct competition between x and y. Otherwise, we say that x and y compete indirectly and we call the edge (x, y) in E(C1,2) an indirect competition between x and y. We have developed a family of digraphs that induce complete components in their (1,2)-step competition graphs that appear to have a minimum number of direct competitions.