The Characterization Of Graphs With Two Kinds Of 2-connected Graphs As Forbidden Minor

For any two graphs G and H,H is called a minor of G,if H can be obtained from G by repeatedly contracting edges,deleting edges and deleting vertices.If there is no minor of G is isomorphic to H,G is called H-minor-free and H is called a forbidden minor of G.The graph minor theory is the basic subject of studying the structural of graphs.In the study of many traditional and difficult problems,it can often be attributed to H-minor-free graphs,such as:the Tutte’s 4-flow-conjecture can be attribute to Petersen-minor-free graphs;Hadwiger’s Conjecture can be attribute to Kn-minor-free graphs.Therefore,it is a diffi-cult problem to characterize graphs which contain no certain classical graphs.In particular,characterizing graphs with a 2-connected graph as a forbidden minor is a difficult research direction,which deserves to be deeply considered.This paper mainly studies the following two problems:(1)Let V8 be a graph constructed from an 8-cycle by connecting the antipodal vertices.There are thirteen 2-connected spanning subgraphs of V8.In particular,one spanning sub-graph is obtained from Petersen graph by deleting two vertices and it is a hard problem to characterize Petersen-minor-free graphs in the academic community.In Chapter 2,we char-acterize internally 4-connected graphs which contain a 2-connected spanning subgraph of V8 as a forbidden minor.(2)Let (?).In Chapter 3,we first prove that if G ∈ R,then G is 3-connected and K2,5-minor-free.Next,we find some special 2-connected graphs and prove that these graphs are K2,5-minor-free.