Removable edges in a 5-connected graph and a construction method of 5-connected graphs Original Research Article

An edge e of a k-connected graph G is said to be a removable edge if image is still k-connected. A k-connected graph G is said to be a quasi image-connected if G has no nontrivial k-separator. The existence of removable edges of 3-connected and 4-connected graphs and some properties of quasi k-connected graphs have been investigated [D.A. Holton, B. Jackson, A. Saito, N.C. Wormale, Removable edges in 3-connected graphs, J. Graph Theory 14(4) (1990) 465–473; H. Jiang, J. Su, Minimum degree of minimally quasi image-connected graphs, J. Math. Study 35 (2002) 187–193; T. Politof, A. Satyanarayana, Minors of quasi 4-connected graphs, Discrete Math. 126 (1994) 245–256; T. Politof, A. Satyanarayana, The structure of quasi 4-connected graphs, Discrete Math. 161 (1996) 217–228; J. Su, The number of removable edges in 3-connected graphs, J. Combin. Theory Ser. B 75(1) (1999) 74–87; J. Yin, Removable edges and constructions of 4-connected graphs, J. Systems Sci. Math. Sci. 19(4) (1999) 434–438]. In this paper, we first investigate the relation between quasi connectivity and removable edges. Based on the relation, the existence of removable edges in k-connected graphs (image) is investigated. It is proved that a 5-connected graph has no removable edge if and only if it is isomorphic to image. For a k-connected graph G such that end vertices of any edge of G have at most image common adjacent vertices, it is also proved that G has a removable edge. Consequently, a recursive construction method of 5-connected graphs is established, that is, any 5-connected graph can be obtained from image by a number of image-operations. We conjecture that, if k is even, a k-connected graph G without removable edge is isomorphic to either image or the graph image obtained from image by removing image disjoint edges, and, if k is odd, G is isomorphic to image.