Self-Contained Graphs

TL;DR

This paper explores the properties of self-contained graphs, introduces new sub-structures, and connects the graph alternative conjecture to its connected version, providing insights into their structural relationships.

Contribution

It defines new sub-structures like removable subgraphs and the foundation, and relates the general graph alternative conjecture to the connected case.

Findings

The general graph alternative conjecture can be deduced from its connected version.

Conditions under which the union of removable subgraphs is also removable are identified.

Examples of self-contained graphs and their sub-structures are presented.

Abstract

A self-contained graph is an infinite graph which is isomorphic to one of its proper induced subgraphs. In this paper, these graphs are studied by presenting some examples and defining some of their sub-structures such as removable subgraphs and the foundation. Then, we show that the general version of graph alternative conjecture, which says every graph has infinitely many strong twins or none, can be deduced from its connected version, which says every connected graph has infinitely many connected strong twins or none. Moreover, we try to find out under what conditions on two arbitrary removable subgraphs, their union is also a removable subgraph.