Self-Contained Graphs

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.

Strong Twins of Ordinary Star-Like Self-Contained Graphs

“…Self-contained graphs are infinite graphs which have isomorphic copies of themselves as proper induced subgraphs. These graphs were studied in [5] and, in this paper, we continue studying them by finding a special kind of self-contained graphs for which a renowned conjecture of Bonato and Tardif [2] comes true.…”

“…To read this paper, we need some few definitions, notations and results we have presented in [5]. Meanwhile, few definitions of infinite graph theory is also needed, all of which can be found in Section 8 of [4].…”

“…For a self-contained graph G, a non-empty proper subgraph H is a removable subgraph if G \ H ∼ = G. Then we write H ∈ Rem(G) and by Iso G (H) we mean the set of all isomorphisms f : G −→ G \ H [5]. We may also need the following two propositions: Proposition 1.1.…”

“…Let G be a self-contained graph, P ∈ Rem(G) and Q be an induced subgraph of G \ P . Then Q ∈ Rem(G \ P ) if and only if P ∪ Q ∈ Rem(G) [5].…”

“…Let G be a self-contained graph and H ∈ Rem(G). Then G contains infinitely many vertex disjoint copies of H [5].…”