Boundary vertices in graphs (Q1869215)

A vertex \(u\) is a boundary vertex of a nontrivial connected graph \(G\) if there is a vertex \(v\) of \(G\) such that (the distance) \(d(w,v)\leq d(u,v)\) for all neighbors \(w\) of \(u\). The subgraph of \(G\) induced by its boundary vertices is the boundary \(\partial (G)\) of \(G\). The authors study the relationship among the periphery Per\((G)\) [\textit{H. Bielak} and \textit{M. M. Syslo}, Stud. Sci. Math. Hung. 18, 269-275 (1983; Zbl 0569.05030)], eccentric subgraph \(\text{Ecc}(G)\) [\textit{G. Chartrand, W. Gu, M. Schultz} and \textit{S. J. Winters}, Networks 34, 115-121 (1999; Zbl 0959.90062)], and boundary \(\partial (G)\) of a connected graph \(G\). Also a characterization of all boundary graphs is established.