子图存在性问题是结构图论的核心问题。本项目主要考虑三方面的问题：k-连通图中拓扑子式和可收缩子图的存在性、电力控制集算法和电力控制数的上界、边染色超图的圈或路划分。其核心内容包括：一、在郁星星等人解决Kelmans-Seymour猜想的方法基础上，研究4-连通图中存在K_5^-拓扑子式及可收缩子图的存在性问题，同时研究3-连通图中存在K_5相关拓扑子式的充分条件；二、围绕Dorbec等人关于k-电力控制数上界的猜想展开研究，希望得到4-正则和一般r-正则图的k-电力控制数好的上界，同时研究电力控制集及相关控制集的算法复杂性以及在弦图子图类上的多项式时间算法；三、研究Bustamante和Stein关于2-边染色一致完全超图的圈划分猜想以及Gyárfás和Sárközy关于非完全超图的线性圈划分猜想，希望在这些圈划分猜想上取得突破性进展。

The existence of subgraph is a core problem of structural graph theory. This project includes the following three topics: the existence of topological minors and contractible subgraphs of k-connected graphs, algorithms of power domination problems and the upper bounds for power domination numbers in graphs, partition edge-colored hypergraphs by cycles or paths. The main content of the project is as follows. (1) Based on Yu's techniques for solving the Kelmans-Seymour Conjecture, we shall research on the existence of K_5^- topological minors and contractible subgraphs in 4-connected graphs. We shall also study the sufficient conditions for 3-connected graphs containing a K_5^- topological minor. (2) Considering the conjecture presented by Dorbec et.al for the upper bound of k-power domination number in regular graphs, we shall give good upper bounds of k-power domination number for 4-regular and general r-regular graphs. We shall also study algorithmic complexity on power domination and other relevant domination problems in graphs, especially study polynomial-time algorithms in subclasses of chordal graphs. (3) We shall study the Bustamante and Stein's cycle-partition conjecture on 2-edge-colored complete hypergraphs, and also study the Gyárfás and Sárközy's loose cycles partition conjecture on non-complete hypergraphs. We hope to make breakthrough on these cycles partition conjectures.