A $(m,n)$-mixed graph is a mixed graph whose edges are assigned one of $m$ colours, and whose arcs are assigned one of $n$ colours. Let $G$ be a $(m,n)$-mixed graph and $\pi=(\alpha,\beta,\gamma_1,\gamma_2,\ldots,\gamma_n)$ be a $(n+2)$-tuple of permutations from $S_m \times S_n \times S_2^n$. We define \emph{switching at a vertex $v$ with respect to $\pi$} as follows. Replace each edge $vw$ of colour $\phi$ by an edge $vw$ of colour $\alpha(\phi)$, and each arc $vx$ of colour $\phi$ by an arc $\gamma_\phi(vx)$ of colour $\beta(\phi)$. In this thesis, we study the complexity of the question: ``Given a $(m,n)$-mixed graph $G$, is there a sequence of switches at vertices of $G$ with respect to the fixed group $\Gamma$ so that the resulting $(m,n)$-mixed graph admits a homomorphism to some $(m,n)$-mixed graph on $2$ vertices?'' We show the following: (1) When restricted to $(m,0)$-mixed graphs $H$ on at most $2$ vertices, the $\Gamma$-switchable homomorphism decision problem is solvable in polynomial time; (2) for each bipartite $(0,n)$-mixed graph $H$, there is a bipartite $(2n,0)$-mixed graph such that the respective $\Gamma$-switchable homomorphism decision problems are polynomially equivalent; (3) For all $(m,n)$-mixed graphs and groups, when $H$ has at most $2$ vertices, the $\Gamma$-switchable homomorphism decision problem is polynomial time solvable; (4) For a yes-instance of the $\Gamma$-switchable homomorphism problem for $(m,0)$-mixed graphs, we can find in quadratic time a sequence of switches on $G$ such that the resulting $(m,0)$-mixed graph admits a homomorphism to $H$. By proving (1)-(4), we show that the $\Gamma$-switchable $2$-colouring problem for $(m,n)$-mixed graphs is solvable in polynomial time for all finite permutation groups $\Gamma$ and provide a step towards a dichotomy theorem for the complexity of the $\Gamma$-switchable homomorphism decision problem. / Graduate

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Research shows that in a system model, when the production rate is adjusted based on the number of items in queue, the nature of the model changes from an open-loop queueing system to a closed-loop feedback control system. Service-rate adjustment can be implemented in a discrete event simulation model, but the effect of this adjustment has not been thoroughly analyzed in the literature. This research considers the design of feedback signals to generate realistic simulation models of production system behavior. A series of simulation experiments is conducted to provide practical guidance for simulation modelers on how adding a service-rate adjustment feedback loop to a queueing system affects system performance.

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In the mid 1900s the area of extremal graph theory took its first propersteps with the proof of Turán’s theorem. In 1963 Pál Erdős asked for an extension of this fundamental result regarding (n, s, q)-graphs; graphs on n vertices in which any s-set of vertices spans at most q edges, and multiple edges are allowed; and raised the question of determining ex(n, s, q), the maximum number of edges spanning such a graph. More recently, Mubayi and Terry looked at the problem of determining the maximum productof the edges in (n, s, q)-graphs. Their proof was further investigated by Day, Falgas-Ravry and Treglown who, in particular, settled a conjecture of Mubayi and Terry regarding the case (s, q) = (4, 6a+3), for a ∈ Z≥2. In this thesis we look at the case (s, q) = (5, 24), which is mentioned as an open problem at the end of the paper by Day, Falgas-Ravry and Treglown. A hypothetical extremal construction was provided by Victor Falgas-Ravry, and we prove it to be asymptotically optimal.

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