The Connectivity Problems Of Two Kinds Of Matroids Related To Bicircular Graphs

Hassler Whitney proposed the conception of matroid in 1935 which aimed at axiomatizing the commonness of various related notions in algebra and graph theory.Because of the ingenious structure of matroids,all the results of the linear indepen-dence not related to certain field can be expressed by matroids.In graph theory,many concepts and properties can also be spread by matroids.The connectivity function of a matroid M is written as λm.We define that AM(X)= rM(X)+ rM(Y)-r(M),where(X,Y)is a partition of E(M).We call(X,Y)is a k-separation of M,if λM(X)≤ k-*1,|X| ≥ k and |Y| ≥ k.A matroid M is n-connected if it has no separations less than n.Biased graphs Ω(G,B)were introduced by Zaslavsky,where G is the underlying graph of Ω.Bicircular graphs Q(G,0)are biased graphs,which contains no balanced cycles.Bicircular frame matroid F(G,θ)and bicircular lift matroid L(G,0)are ma-troids defined on the ground edge set of bicircular graphs.In this paper,we study the connectivity of these two kinds of matroids and their related bicircular graphs.We obtain that for any k ≥ 4,except for several special cases,F(G,0)and L(G,0)are k-connected if and only if Ω(G,0)is(k-1)-connected.We organize this paper as follows.In Chapter 1,we introduce some basic defini-tions and related conclusions.In Chapter 2,we prove that for any k ≥ 4 and except for several special cases,F(G,0)is k-connected if and only if Ω(G,θ)is(k-1).connected.We make several special cases as propositions.We prove by contradiction.In Chapter 3,we use a similar method to prove that for any k>4 and except for sev-eral special cases,L(G,θ)is k-connected if and only if Ω(G,θ)is(k-1)-connected.In Chapter 4,some further research work is given.