On maximum critically h-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...

On Critically Perfect Graphs on Critically Perfect Graphs

A perfect graph is critical if the deletion of any edge results in an imperfect graph. We give examples of such graphs and prove some basic properties. We relate critically perfect graphs to well-known classes of perfect graphs, investigate the structure of the class of critically perfect graphs, and study operations preserving critical perfectness.

Determination and Construction of Critically 2-Connected Graphs