Critically $n$-connected graphs

Critically «connected Graphs

The following result is proved. Every «-connected graph contains either a vertex whose removal results in a graph which is also «-connected or a vertex of degree less than (3n—1)/2. Introduction. A graph G is said to be n-connected if the removal of fewer than « vertices from G neither disconnects it nor reduces it to the trivial graph consisting of a single vertex. The maximum value of « for w...

On critically k-edge-connected graphs

Let G be a simple graph on n vertices having edge-connectivity /(.' (G) > a and minimum degree o(G) We say G is k-critical if /(.' (G) = k and /(.' (G e) < k for every edge e of G. In this paper we prove that a k-critical graph has 1<' (G) o(G). We descri be a number of classes of k-cri tical graphs and consider the problem of determining the edge-maximal ones.

On Determination and Construction of Critically 2-Connected Graphs∗

Kriesell proved that every almost critical graph of connectivity 2 nonisomorphic to a cycle has at least 2 removable ears of length greater than 2. We improve this lower bound on the number of removable ears. A necessary condition for critically 2-connected graphs in terms of a forbidden minor is obtained. Further, we investigate properties of a special class of critically 2-connected series-pa...

Determination and Construction of Critically 2-Connected Graphs

The characterization of edge-maximal critically k-edge connected graphs

Let G be a simple graph on n vertices having edge-connectivity 1(' (G) > O. We say G is k-critical if 1(' (G) :::: k and 1(1 (G-e) < k for every edge e of G. We denote by ~(n,k) the set of all k-critical graphs on n vertices. In this paper we prove that the maximum number of edges of a graph G in ~(n,k) to be: k(n-k) if n ~ 3k; and L ~ (n+k)2 J. if k + 1 s n < 3k. Further, we characterise the e...