Constructing Almost Peripheral and Almost Self-centered Graphs Revisited

Abstract

The center and the periphery of a graph is the set of vertices with minimum resp. maximum eccentricity in it. A graph is almost self-centered (ASC) if it contains exactly two non-central vertices and is almost peripheral (AP) if all its vertices but one lie in the periphery. Answering a question from (Taiwanese J. Math. 18 (2014), 463--471) it is proved that for any integer $r \geq 1$ there exists an $r$-AP graph of order $4r-1$. Using this result it is proved that any graph $G$ can be embedded into an $r$-AP graph by adding at most $4r-2$ vertices to $G$. A construction of ASC graphs from (Acta Math. Sin. (Engl. Ser.) 27 (2011), 2343--2350) is corrected and refined. Two new constructions of ASC graphs are also presented. Strong product graphs that are AP graphs are also characterized and it is shown that there are no strong product graphs that are ASC graphs. We conclude with some related open problems.