Gamma-Switchable 2-Colourings of (m,n)-Mixed Graphs
| dc.contributor.author | Kidner, Arnott | |
| dc.contributor.supervisor | MacGillivray, Gary | |
| dc.contributor.supervisor | Brewster, Richard Charles | |
| dc.date.accessioned | 2021-08-31T17:15:58Z | |
| dc.date.available | 2021-08-31T17:15:58Z | |
| dc.date.copyright | 2021 | en_US |
| dc.date.issued | 2021-08-31 | |
| dc.degree.department | Department of Mathematics and Statistics | |
| dc.degree.level | Master of Science M.Sc. | en_US |
| dc.description.abstract | 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. | en_US |
| dc.description.scholarlevel | Graduate | en_US |
| dc.identifier.uri | http://hdl.handle.net/1828/13337 | |
| dc.language | English | eng |
| dc.language.iso | en | en_US |
| dc.rights | Available to the World Wide Web | en_US |
| dc.subject | Discrete Mathematics | en_US |
| dc.subject | Graph Theory | en_US |
| dc.subject | Graph Homomorphisms | en_US |
| dc.title | Gamma-Switchable 2-Colourings of (m,n)-Mixed Graphs | en_US |
| dc.type | Thesis | en_US |