I was studying about Almost self-centered graphs.
My doubt is what would be the minimum number of vertices for such graphs.
My idea: I think its 4 and the graph that satisfy this condition is $P_4$ where end vertices are not in the center of $P_4$. Is my solution correct? If not, then kindly give hints or suggestions, thanks.
Definition : Almost self-centered (ASC) graphs are introduced as the graphs with exactly two non-central vertices.
NOTE : there is another class of graphs known as almost peripheral graphs. Almost peripheral (AP) graphs are introduced as graphs G with |P(G)| = |V (G)|−1 (and |C(G)| = 1). I think $P_3$ is AP graph