| For two given graphs H and G,if H can be obtained from a subgraph of G by contracting edges then deleting the resulting loops and parallel edges,we call H a minor of G.If G has no minor isomorphic to H,G is H-minor-free,and H is a forbidden minor of G.There are two graphs that obtained from the Octahedron(denoted by Oct)by 3-splitting a vertex,we call the planar one Oct1+,and the non-planar one Oct2+.Suppose K2,5 has a partition(｛a1,a2｝;｛b1,b2,b3,b4,b5｝),let K1,1,5 be the graph obtained from K2,5 by adding an edge a1a2.In graph theory,there are many important conjectures about H-minor-free graphs such as Hadwiger’s conjecture and Tutte’s 4-flow conjecture.To solve these conjectures,we are focus on the structures of K6-minor-free graphs and Petersen-minor-free graphs now.Since they are 3-connected graphs with 15 edges,many scholars try to characterize every 3-connected graph with edges less than 15 to get close to K6 and Petersen graph.The two graphs Oct1+ and Oct2+ in this paper are 3-connected graphs with 13 edges that have not been characterized.In addition,characterizing graph classes exclude a 2-connected graph as a minor is also a hot spot.And usually these graph classes show good properties on traversability(such as hamiltonicity).According to the researches of K2,4-minor-free graphs,K1,1,4-minor-free graphs and K2,5minor-free graphs,we explore the structure of 4-connected planar K1,1,5-minor-free graphs in this paper.The chapters of this paper are as follows:1.In chapter 1,we introduced the basic knowledge,research backgrounds and the research contents of this paper.2.In chapter 2,we characterized 4-connected Oct1+-minor-free graphs and 4-connected Oct2+-minor-free graphs.We obtained a necessary condition for a 4-connected graph to be Oct1+-minor-free or Oct2+-minor-free.For the planar graph Oct1+,we completely characterized all planar Oct1+-minor-free graphs.3.In chapter 3,we first proved that there is no vertex of degree more than five in a 4-connected planar K1,1,5-minor-free graph G.Thus for any vertex of G,the degree is 4 or 5.We explored the structure of graph G when G has two adjacent vertices of degree 5,we proved the two 5-degree vertices exactly have two common neighbors,and characterized G with vertices less than 13.When the vertices of G equal to or more than 13,we give some structural characterizations of G.4.In chapter 4,we summarized the main work in this paper,analyzed the problems and shortcomings of our research,and planed the future research direction. |
