On Kernels And Starshaped Sets Under F-convexity

Let F be a family of sets in Rd,M(?)Rd.If for any two distinct points x,y?M,there exists a set F ?F such that x,y?F and F(?)M,then M is called F-convex.If there exists a point x ?M such that for any point y?M,there is a set F?J such that x,y?F and F(?)M,,then M is called a starshaped set according to x under F-convexity.The set K of all such points x is the kernel of M under F-convexity.In this thesis we discuss F-convexity in lattice graphs.By selecting F as a family of sets consisting of some paths that satisfying some certain properties,we define the s-convexity and t-convexity about the vertex set U in the regular hexagonal lattice graph(63)and regular triangular lattice graph(36)respectively,and we also introduce the con-cepts of r-starshaped set and r-kernel respectively.Furthermore,we study the properties of the vertex set U under the given convexity,and describe the minimal r-starshaped set(rm-starshaped set)and r-kernel(rm-kernel)of the vertex set U in the lattice graph(63)and(36)respectively.