A vertex v of a graph G is a boundary vertex if there exists a vertex u such that the distance in G from u to v is at least the distance from u to any neighbour of v. We give the best possible lower bound, up to a constant factor, on the number of boundary vertices of a graph in terms of its minimum degree (or maximum degree). This settles a problem introduced by Hasegawa and Saito.

Let P be an H-polytope in R with vertex set V . The vertex centroid is defined as the average of the vertices in V . We prove that computing the vertex centroid of an H-polytope is #P-hard. Moreover, we show that even just checking whether the vertex centroid lies in a given halfspace is already #P-hard for H-polytopes. We also consider the problem of approximating the vertex centroid by findin...

the wiener index w(g) of a connected graph g is defined as the sum of the distances betweenall unordered pairs of vertices of g. the eccentricity of a vertex v in g is the distance to avertex farthest from v. in this paper we obtain the wiener index of a graph in terms ofeccentricities. further we extend these results to the self-centered graphs.

A graph is distance-hereditary if for any pair of vertices, their distance in every connected induced subgraph containing both vertices is the same as their distance in the original graph. Distance hereditary graphs are exactly the graphs with rank-width at most 1. The Distance-Hereditary Vertex Deletion problem asks, given a graph G on n vertices and an integer k, whether there is a set S of a...

For a family of graphs $$\mathcal {F}$$ , Weighted -Deletion is the problem for which input vertex weighted graph $$G = (V, E)$$ and goal to delete $$S \subseteq V$$ with minimum weight such that \setminus S \in \mathcal . Designing constant-factor approximation algorithm large subclasses perfect has been an interesting research direction. Block graphs, 3-leaf power interval are known admit alg...

Distance labeling is a preprocessing technique introduced by Peleg [Journal of Graph Theory, 33(3)] to speed up distance queries in large networks. Herein, each vertex receives a (short) label and, the distance between two vertices can be inferred from their two labels. One such preprocessing problem occurs in the hub labeling algorithm [Abraham et al., SODA’10]: the label of a vertex v is a se...

Let Γ be a distance-regular graph with diameter d. For vertices x and y of Γ at distance i, 1 ≤ i ≤ d, we define the sets Ci(x, y) = Γi−1(x) ∩ Γ(y), Ai(x, y) = Γi(x) ∩ Γ(y) and Bi(x, y) = Γi+1(x) ∩ Γ(y). Then we say Γ has the CABj property, if the partition CABi(x, y) = {Ci(x, y), Ai(x, y), Bi(x, y)} of the local graph of y is equitable for each pair of vertices x and y of Γ at distance i ≤ j. ...