## Games, Graphs, and Groups

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## Abstract

This dissertation contains various combinatorial results about games, graphs and finite Abelian groups.

A linear configuration is said to be *common* in an Abelian group

We call a graph *strongly common* if for every colouring

A set *sum-free* if it contains no elements

We say that a family *linear graph code* if the symmetric difference of the edge sets of any two graphs in

The game *cops and robbers* is played on a graph *catch* the robber piece. In Chapter 6, we present a new algorithm that determines for any

In the *domination game*, two players called Dominator and Staller select vertices in a graph *dominated* if it has been selected or is adjacent to a selected vertex. Each selected vertex must dominate a new vertex, and the game ends once every vertex in *total domination*, then Dominator has a strategy to end the game in