Geodeticity of the contour of chordal graphs
Discrete Applied Mathematics
Discrete Applied Mathematics
| Hi-index | 0.06 |
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 per 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.