## All Dissertations

5-2014

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Legacy Department

Mathematical Science

Goddard, Wayne

Rall, Douglas

#### Committee Member

Hedetniemi, Stephen

Brown, James

#### Committee Member

Matthews, Gretchen

#### Abstract

An identifying code in a graph is a dominating set that also has the property that the closed neighborhood of each vertex in the graph has a distinct intersection with the set. The minimum cardinality of an identifying code in a graph $G$ is denoted $\gid(G)$. We consider identifying codes of the direct product $K_n \times K_m$. In particular, we answer a question of Klav\v{z}ar and show the exact value of $\gid(K_n \times K_m)$. It was recently shown by Gravier, Moncel and Semri that for the Cartesian product of two same-sized cliques $\gid(K_n \Box K_n) = \lfloor{\frac{3n}{2}\rfloor}$. Letting $m \ge n \ge 2$ be any integers, we show that $\IDCODE(K_n \cartprod K_m) = \max\{2m-n, m + \lfloor n/2 \rfloor\}$. Furthermore, we improve upon the bounds for $\IDCODE(G \cartprod K_m)$ and explore the specific case when $G$ is the Cartesian product of multiple cliques. Given two disjoint copies of a graph $G$, denoted $G^1$ and $G^2$, and a permutation $\pi$ of $V(G)$, the permutation graph $\pi G$ is constructed by joining $u \in V(G^1)$ to $\pi(u) \in V(G^2)$ for all $u \in V(G^1)$. The graph $G$ is said to be a universal fixer if the domination number of $\pi G$ is equal to the domination number of $G$ for all $\pi$ of $V(G)$. In 1999 it was conjectured that the only universal fixers are the edgeless graphs. We prove the conjecture.

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