I have successfully defended my master’s thesis on graph-transverse matching problems! It considers the computational complexity of deciding whether a given graph admits a matching which covers every copy of a fixed tree or cycle.
The thesis is related to my previous work on cycle-transverse matchings and P4-transverse matchings and, roughly speaking, shows that H-transverse matchings are NP-hard to find when H is a big cycle or tree, and tractable when H is a triangle or a small tree.
Given graphs G, H, is it possible to find a matching which, when deleted from G, destroys all copies of H? The answer is obvious for some inputs—notably, when G is a large complete graph the answer is “no” — but in general this can be a very difficult question. In this thesis, we study this decision problem when H is a fixed tree or cycle; our aim is to identify those H for which it can be solved efficiently.
The H-transverse matching problem, TM(H) for short, asks whether an input graph admits a matching M such that no subgraph of G−M is isomorphic to H. The main goal of this thesis is the following dichotomy. When H is a triangle or one of a few small-diameter trees, there is a polynomial-time algorithm to find an H-transverse matching if one exists. However, TM(H) is NP-complete when H is any longer cycle or a tree of diameter ≥ 4. In addition, we study the restriction of these problems to structured graph classes.
I am grateful to NSERC for funding my degree with a Alexander Graham Bell Canada Graduate Scholarship, and to my supervisor Jing Huang.