Maximal and Minimal Vertex-Critical Graphs of Diameter Two

Maximal and Minimal Vertex - critical Graphsof Diameter

A graph is vertex-critical (edge-critical) if deleting any vertex (edge) increases its diameter. A conjecture of Simon and Murty stated thatèvery edge-critical graph of diameter two on vertices contains at most 1 4 2 edges'. This conjecture has been established for suuciently large. For vertex-critical graphs, little is known about the number of edges. Plesn ik implicitly asked whether it is al...

Regular vertex diameter critical graphs

A graph is called vertex diameter critical if its diameter increases when any vertex is removed. Regular vertex diameter critical graphs of every valency k ≥ 2 and diameter d ≥ 2 exist, raising the question of identifying the smallest such graphs. We describe an infinite family of k-regular vertex diameter critical graphs of diameter d with at most kd+ (2k − 3) vertices. This improves the previ...

Minimum vertex-diameter-2-critical graphs

minimal, vertex minimal and commonality minimal cn-dominating graphs

we define minimal cn-dominating graph $mathbf {mcn}(g)$, commonality minimal cn-dominating graph $mathbf {cmcn}(g)$ and vertex minimal cn-dominating graph $mathbf {m_{v}cn}(g)$, characterizations are given for graph $g$ for which the newly defined graphs are connected. further serval new results are developed relating to these graphs.

On the diameter of total domination vertex critical graphs

In this paper, we consider various types of domination vertex critical graphs, including total domination vertex critical graphs and independent domination vertex critical graphs and connected domination vertex critical graphs. We provide upper bounds on the diameter of them, two of which are sharp. MSC (2010): 05C12, 05C69