# Self-Contained Graphs

@article{Shekarriz2015SelfContainedG, title={Self-Contained Graphs}, author={Mohammad Hadi Shekarriz and Madjid Mirzavaziri}, journal={arXiv: Combinatorics}, year={2015} }

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…

## 4 Citations

Strong Twins of Ordinary Star-Like Self-Contained Graphs

- Mathematics
- 2015

A self-contained graph is an infinite graph which is isomorphic to one of its proper induced subgraphs. In this paper, ordinary star-like self-contained graphs are introduced and it is shown that…

Siblings of countable cographs

- Mathematics
- 2020

We show that every countable cograph has either one or infinitely many siblings. This answers, very partially, a conjecture of Thomasse. The main tools are the notion of well quasi ordering and the…

Infinite Ramsey-minimal graphs for star forests

- Mathematics
- 2021

For graphs F , G, and H , we write F → (G,H) if every red-blue coloring of the edges of F produces a red copy of G or a blue copy of H . The graph F is said to be (G,H)-minimal if it is…

On Ramsey-Minimal Infinite Graphs

- Mathematics, Computer ScienceElectron. J. Comb.
- 2021

This work proves some compactness results relating this problem to the finite case, then gives some general conditions for a pair $(G,H)$ to have a Ramsey-minimal graph, and proves, for example, that if $G=S_\infty$ is an infinite star and $H=nK_2$, $n \geqslant 1$ is a matching, then the pair admits no Ramsey-Minimal graphs.

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