Graphs With Parity Conditions On Distances Between Cut And Noncut Vertices

In finite graphs, a cut vertex is a vertex whose removal increases the number of connected components. We study five classes of connected graphs that satisfy parity conditions on distances between cut and noncut vertices. We provide characterizations for the block graphs of these graphs and describe the structure of the set of all cut vertices within them. Additionally, we present structural characterizations of these graph classes and establish comprehensive criteria for all possible intersections among the five mentioned classes.