# User:Lschumac

## Minimum i, k Values for Complete (i,k)-Step Competition Graphs Research

**Week 1**

*Monday May 26*

- Memorial Day

*Tuesday May 27*

-Orientation

-Researched background in Food Webs and Competition Graphs

*Wednesday May 28*

-Met with Dr. Factor

-Created (1,k)-step competition graphs for food webs in BioMath

-Identified similarities between webs with and without complete competition graphs (# of vertices vs. # of arcs, # of chains in the web, in and out degree of vertices, etc.)

*Thursday May 29*

- Determined why complete competition graphs cannot be created for specific food webs

- Created and analyzed the minimum k value required to find a complete (1, k) step competition graph for acyclic digraphs (with 8 vertices) in which all vertices share the same sum of indegree and outdegree

*Friday May 30*

- Determined why complete competition graphs cannot be created for specific food webs

- Created and analyzed the minimum k value required to find a complete (1, k) step competition graph for acyclic digraphs (with 10 and 12 vertices) in which all vertices share the same sum of indegree and outdegree

**Week 2**

*Monday June 2*

- Looked at the relationship between the minimum i,k values needed for a complete competition graph to the length of the shortest chains from each vertex to the primary producer (once again with graph in which each vertex has the same sum of indegree and outdegree)

*Tuesday June 3*

- Found examples of graphs to back up the patterns I have been seeing in digraphs in which the sum of the in and out degree is the same for all vertices

*Wednesday June 4*

- Finalized patterns I have found and gathered examples to be able to explain findings to Zornica

- Collaborated with Zornica to discuss progress and findings

- Reported our progress to Dr. Factor

*Thursday June 5*

- Found a few digraphs which do not seem to fit with my theory

- Tried to find differences between digraphs in which the patterns do occur versus the digraphs in which the patterns do not occur

*Friday June 6*

- Began researching the history of competition graphs in food webs for the background of my final paper

**Week 3**

*Monday June 9*

- Continued to look for a pattern between minimum i, k values and the longest and shortest paths of each vertex to a primary producer of an acyclic digraph in which the sum of the in and out degrees are the same for each vertex.

- Found patterns between the average shortest path with the minimum i value and the average longest path with the minimum k value.