5 Further Studies

Kotzig, A. S uvislost' a pravidelnn a s uvislost' kone cn yh grafov. maintenance of the four-connected components of a graph. Proc. A fast algorithm for optimally increasing the edge-connectivity. Figure 9: A 4-edge-connected graph and 3-edge{cuts in its 3-vertex-components for incremental maintenance of this structure is suggested. No general l-cuts description, l k + 1, is known. To item 4. The system of k-vertex-connected classes of a graph is, in general, thinner than one of k-edge-connected classes. But we show in Fig. 9 that there can be 3-edge-cuts of 3-vertex-connected components which do not generate any 3-edge-cuts of G in any way. Therefore, we cannot achieve an immediate connection between k-vertex-connected components, k 3, and edge-cuts (such as we have for 3-edge-connected components). To item 5. A full classiication is found for the case, where k = 3 and the cactus model is a path. 3. To develop a deenition of a k-component, k 4, adequate to the k-cuts/(k + 1)-classes investigation ((rstly, for k = c(G) + 1). Using this deenition: to give a description l-cuts from a xed k-class-family, l k; to give a description l-cuts from C l k;2 relative to the k-components system structure, l k, in particular, as combined from k-component cuts of less or equal cardinalities. 4. To develop an edge l-cuts description related to the structure of the system of k-vertex-connected components of a graph, l k, k 3. 5. To describe the system of minimum cuts of a graph G, c(G) 3, as edge-sets (at present, we have only the indirect description of these cuts, as corresponding to vertex partitions given by the cactus model). It is desirable that descriptions developed be graph invariants. Following is an information concerning the current state of studies. To item 1. An incremental maintenance algorithm for the case of inserting edges is presented in D93], W93] (the same one, invented in parallel). To item 2. Seems to be done, in general. A cut from C l 3;2 is represented by the set of all its 3-class-induced cuts C S of less cardinality, for all 3-classes S broken by C (see Th. 4). The analysis of the system of these cuts implies that the relevant nodes and cycles of the 3-quotient appear to deene a subtree-of-cycles with either one, or two articulation nodes incident to each its cycle. To item 3. There is a natural way …