Embeddings into almost self-centered graphs of given radius

Abstract

A graph is almost self-centered (ASC) if all but two of its vertices are central. An almost self-centered graph with radius r is called an r-ASC graph. The r-ASC index θ r (G) of a graph G is the minimum number of vertices needed to be added to G such that an r-ASC graph is obtained that contains G as an induced subgraph. It is proved that θ r (G) ≤ 2r holds for any graph G and any r ≥ 2 which improves the earlier known bound θ r (G) ≤ 2r +1. It is further proved that θ r (G) ≤ 2r − 1 holds if r ≥ 3 and G is of order at least 2. The 3-ASC index of complete graphs is determined. It is proved that θ 3 (G) ∈ {3, 4} if G has diameter 2 and for several classes of graphs of diameter 2 the exact value of the 3-ASC index is obtained. For instance, if a graph G of diameter 2 does not contain a diametrical triple, then θ 3 (G) = 4. The 3-ASC index of paths of order n ≥ 1, cycles of order n ≥ 3, and trees of order n ≥ 10 and diameter n−2 are also determined, respectively, and several open problems proposed.

Characterization of general position sets and its applications to cographs and bipartite graphs

“…Graphs of diameter 2 form one of the most interesting classes of graph theory, after all, as it is well-known, almost all graphs have diameter 2. They are still extensively investigated, the papers [2,3,22] are examples of recent developments on these graphs. In this section we are going to use Theorem 3.1 in the case of graphs of diameter 2.…”

On Almost Self-centered Graphs and Almost Peripheral Graphs

“…Whenever the extremal graphs have a neat form, we also describe them. For related research, see [4][5][6]11].…”

Comparison of Wiener index and Zagreb eccentricity indices

“…If G = (V (G), E(G)) is a graph, we will use n(G) = |V (G)| for its order and m(G) = |E(G)| for its size. The degree deg G (v) of v ∈ V (G) is the number of vertices in G adjacent to v. The complement of G is denoted with G. The eccentricity ε G (v) (or ε(v) for short) of a vertex v ∈ V (G) is the maximum distance from v to the vertices of G, that is, ε G [31], and the total eccentricity…”