On F-Convexity In Tiling Graphs

Let F be a family of sets in Rd,where d? 2.A set M(?)Rd is an F-convex set if for any two different points x,y?M,there exists a set F ? F such that x,y?F and F(?)M.At present,the study on F-convexity mainly focuses on the properties of F-convex sets when F is a family of connected sets with some geometric structure,or a family of discrete point sets with certain geometric structure,or a family of some subgraphs with certain property.Meanwhile,the properties of starshaped sets and their kernels under the frame of F-convexity are also investigated.In Chapter 1,we introduce the research background and the research significance of this thesis,and present our main results.Some necessary definitions and notation are given at the end of this chapter.In Chapter 2,we define a set to be a poidge-convex set,a special F-convex set when F is a non-connected set with certain geometric structure.On one hand,we study the poidge-convexity of starshaped sets and non-simply connected sets,and characterize symmetric starshaped sets with poidge-convexity,and give a sufficient condition for or-dinary one-sided starshaped set and some non-simply connected set to be poidge-convex,respectively.On the other hand,we prove the convex surfaces of extraordinary con-vex polyhedron and right cone are poidge-convex and characterize all tetrahedrons with poidge-convex surface.In Chapter 3,we generalize the definition of poidge-convexity to tiling graphs,and characterize horizontally convex paths,horizontally convex grid graphs and general grid graphs with poidge-convexity in square tiling graphs,respectively.We also characterize the poidge-convex path and give some properties of poidge-convex grid graphs with convex boundary in triangular tiling graphs.In Chapter 4,we choose that F is a family of paths with some properties.On one hand,we study the r-kernel of finite t-convex subgraph of infinite triangular tiling graphs,and give the optimal bound of the smallest integers rm such that the r-kernel under the t-convexity is non-empty.On the other hand,we study the r-kernel of finite s-convex subgraph of infinite regular hexagonal tiling graphs,and give the optimal bound of the smallest integers rm such that the r-kernel under the s-convexity is non-empty.