Boundary vertices in graphs

Abstract

The distance d( u, v) between two vertices u and v in a nontrivial connected graph G is the length of a shortest u– v path in G. For a vertex v of G, the eccentricity e( v) is the distance between v and a vertex farthest from v. A vertex v of G is a peripheral vertex if e( v) is the diameter of G. The subgraph of G induced by its peripheral vertices is the periphery Per( G) of G. A vertex u of G is an eccentric vertex of a vertex v if d( u, v)= e( v). A vertex x is an eccentric vertex of G if x is an eccentric vertex of some vertex of G. The subgraph of G induced by its eccentric vertices is the eccentric subgraph Ecc( G) of G. A vertex u of G is a boundary vertex of a vertex v if d( w, v)⩽ d( u, v) for all w∈ N( u). A vertex u is a boundary vertex of G if u is a boundary vertex of some vertex of G. The subgraph of G induced by its boundary vertices is the boundary ∂( G) of G. A graph H is a boundary graph if H= ∂( G) for some graph G. We study the relationship among the periphery, eccentric subgraph, and boundary of a connected graph and establish a characterization of all boundary graphs. It is shown that for each triple a, b, c of integers with 2⩽ a⩽ b⩽ c, there is a connected graph G such that Per( G) has order a, Ecc( G) has order b, and ∂( G) has order c. Moreover, for each triple r, s, t of rational numbers with 0< r⩽ s⩽ t⩽1, there is a connected graph G of order n such that | V(Per( G))|/ n= r, | V(Ecc( G))|/ n= s, and | V( ∂( G))|/ n= t.