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Title: | Dominating sets in graphs with no long induced paths |

Authors: | Shi, Jessica |

Advisors: | Liu, Chun-Hung Chudnovsky, Maria |

Department: | Mathematics |

Certificate Program: | Applications of Computing Program |

Class Year: | 2018 |

Abstract: | 3-coloring is a classically difficult problem, and as such, it is of interest to consider the computational complexity of 3-coloring restricted to certain classes of graphs. \(P_t\)-free graphs are of particular interest, and the problem of 3-coloring \(P_8\)-free graphs remains open. One way to prove that 3-coloring graph class \(\mathcal{G}\) is polynomial is by showing that for all \(G \in \mathcal{G}\), there exists a constant bounded dominating set in \(G\); that is to \(G\) contains a dominating set \(S\) such that \(|S| \leq K_\mathcal{G}\) for constant \(K_\mathcal{G}\). In this paper, we prove that there exist constant bounded dominating sets in subclasses of \(P_t\)-free graphs. Specifically, we prove that excepting certain reducible configurations which can be disregarded in the context of 3-coloring, there exist constant bounded dominating sets in \(\{P_6, \textrm{triangle}\}\)-free and \(\{P_7, \textrm{triangle}\}\)-free graphs. We also provide a semi-automatic proof for the latter case, due to the algorithmic nature of the proof. |

URI: | http://arks.princeton.edu/ark:/88435/dsp01jw827f42s |

Type of Material: | Princeton University Senior Theses |

Language: | en |

Appears in Collections: | Mathematics, 1934-2020 |

Files in This Item:

File | Description | Size | Format | |
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SHI-JESSICA-THESIS.pdf | 366.51 kB | Adobe PDF | Request a copy |

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