Some generalized theorems on connectivity

R. E. Nettleton
1960 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
The "£-dense" subgraphs of a connected graph G are connected and contain neighbours of all but at most k-l points. We consider necessary and sufficient conditions that a point be in I\, the union of the minimal &-dense subgraphs. It is shown that T k contains all the u [m, &]-isthmuses" and u [m, ^-articulators"-minimal subgraphs which disconnect the graph into at least k +1 disjoint graphs-and that an [m, &]-isthmus or [m, &]-articulator of T k disconnects G. We define "central points,"
more » ... " of a point, and "chromatic number" and examine the relationship of these concepts to connectivity. Many theorems contain theorems previously proved (1) as special cases. 1. Definitions. The concepts points, graph, and subgraph will be used here in precisely the same sense as in a previous paper (1), in which were also defined the union, intersection, and difference of two subgraphs, together with neighbours, path of length k, diameter of a graph, connected points and graphs, m-connected and completely connected graphs, articulator, a subgraph which disconnects G, and the partition of a disconnected graph. Unless otherwise specified, a connected graph G will have a finite number ' V of points, and the null graph will be assumed disconnected. The distance between a point pi and a subgraph 5 contained in G-pi is the smallest positive integer a such that for some point p 2 in 5 there is a path of length a connecting p\ and pi. A subgraph G' disconnects two subgraphs which are contained in separate graphs of the partition of G-G'. An [m, k]-isthmus ([m, k]-articulator) is a completely connected (not completely connected) subgraph G' which disconnects G and has precisely m points, such that G' contains no proper disconnecting subgraph, and the partition of G-G' consists of at least k +1 graphs. The generic term isthmus will refer to a subgraph which, for any m, is an [m, l]-isthmus. Q fc will denote the union of all subgraphs which, for any m, are [m, ^-articulators or [m, &]-isthmuses of G. In Figure I , the points {2, 5} determine a [2, l]-isthmus, while {2, 4} determine a [2, 2]articulator.
doi:10.4153/cjm-1960-048-9 fatcat:bdhqjjz7rnevdfr3g43xgsjufy