## Poster Session #1: UC Ballroom

#### Presentation Type

Poster

#### Faculty Mentor’s Full Name

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 *C _{1,2}(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

*d*and

_{D\{x}}(y, z) = 1*d*or,

_{D\{y}}(x, z) =< 2*d*and

_{D\{y}}(x, z) = 1*d*. We extend this definition and say that given

_{D\{x}}(y, z) =< 2*x, y*in

*V(D)*such that

*(x, y)*in

*E(C*,

_{1,2})*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(C*a direct competition between

_{1,2})*x*and

*y*. Otherwise, we say that x and y compete indirectly and we call the edge

*(x, y)*in

*E(C*an indirect competition between

_{1,2})*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

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 *C _{1,2}(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

*d*and

_{D\{x}}(y, z) = 1*d*or,

_{D\{y}}(x, z) =< 2*d*and

_{D\{y}}(x, z) = 1*d*. We extend this definition and say that given

_{D\{x}}(y, z) =< 2*x, y*in

*V(D)*such that

*(x, y)*in

*E(C*,

_{1,2})*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(C*a direct competition between

_{1,2})*x*and

*y*. Otherwise, we say that x and y compete indirectly and we call the edge

*(x, y)*in

*E(C*an indirect competition between

_{1,2})*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.