On the metric subgraphs of a graph

TL;DR

This paper investigates the existence and properties of graphs whose metric subgraphs—center, annulus, and periphery—are all paths, cycles, or regular graphs, providing specific existence conditions and open problems.

Contribution

It characterizes when graphs with specific types of metric subgraphs exist and determines their possible orders, advancing understanding of graph metric structures.

Findings

Graphs with all path metric subgraphs exist for n ≥ 13.

Graphs with all cycle metric subgraphs exist for n ≥ 15.

Conditions for graphs with regular metric subgraphs are established.

Abstract

The three subgraphs of a connected graph induced by the center, annulus and periphery are called its metric subgraphs. The main results are as follows. (1) There exists a graph of order $n$ whose metric subgraphs are all paths if and only if $n\ge 13$ and the smallest size of such a graph of order $13$ is $22;$ (2) there exists a graph of order $n$ whose metric subgraphs are all cycles if and only if $n\ge 15,$ and there are exactly three such graphs of order $15;$ (3) for every integer $k\ge 3,$ we determine the possible orders for the existence of a graph whose metric subgraphs are all connected $k$-regular graphs; (4) there exists a graph of order $n$ whose metric subgraphs are connected and pairwise isomorphic if and only if $n\ge 24$ and $n$ is divisible by $3.$ An unsolved problem is posed.