Embeddings into almost self-centered graphs of given radius

TL;DR

This paper investigates the minimal number of vertices needed to embed a given graph into an almost self-centered graph of a specified radius, improving bounds and determining exact indices for various graph classes.

Contribution

It establishes tighter bounds for the $r$-ASC index and determines exact values for specific graph classes, advancing understanding of embeddings into almost self-centered graphs.

Findings

Proved $ heta_r(G)\le 2r$ for all graphs and $r extgreater 1$

Established $ heta_r(G)\le 2r-1$ for graphs with $r extgreater 2$ and order at least 2

Determined the $3$-ASC index for complete graphs, paths, cycles, and certain trees.

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 $\theta_r(G)$ 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 $\theta_r(G)\le 2r$ holds for any graph $G$ and any $r\ge 2$ which improves the earlier known bound $\theta_r(G)\le 2r+1$. It is further proved that $\theta_r(G)\le 2r-1$ holds if $r\geq 3$ and $G$ is of order at least $2$. The $3$-ASC index of complete graphs is determined. It is proved that $\theta_3(G)\in \{3,4\}$ 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 $\theta_3(G) = 4$. The $3$-ASC index of paths of order $n\geq 1$, cycles of order $n\geq 3$, and trees of order $n\geq 10$ and diameter $n-2$ are also determined, respectively, and several open problems proposed.