• Corpus ID: 119155324

Self-Contained Graphs

  title={Self-Contained Graphs},
  author={Mohammad Hadi Shekarriz and Madjid Mirzavaziri},
  journal={arXiv: Combinatorics},
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… 
Strong Twins of Ordinary Star-Like Self-Contained Graphs
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
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
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
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.


Large families of mutually embeddable vertex-transitive graphs
For each infinite cardinal, this work considers examples of 2 many nonisomorphic vertex-transitive graphs of order that are pairwise isomorphic to induced subgraphs of each other that are also universal.
Mutually embeddable graphs and the tree alternative conjecture
We prove that if a rayless tree T is mutually embeddable and non-isomorphic with another rayless tree, then T is mutually embeddable and non-isomorphic with infinitely many rayless trees. The proof
A proof of the rooted tree alternative conjecture
The analogue of the conjecture that the number of isomorphism classes of trees mutually embeddable with a given tree T is either 1 or infinite is proved, which is the analogue of their conjecture for rooted trees.
Twins of rayless graphs
It is proved that a rayless graph has either infinitely many twins or none, and two non-isomorphic graphs are twins if each is isomorphic to a subgraph of the other.
Invariant subsets of scattered trees. An application to the tree alternative property of Bonato and Tardif
A tree is scattered if no subdivision of the complete binary tree is a subtree. Building on results of Halin, Polat and Sabidussi, we identify four types of subtrees of a scattered tree and a
The Random Graph
  • P. Cameron
  • Computer Science, Mathematics
    The Mathematics of Paul Erdős II
  • 2013
The paradoxical result that there is a unique (and highly symmetric) countably infinite random graph, and its automorphism group, form the subject of the present survey.
Handbook of product graphs, 2nd edition
  • CRC Press,
  • 2011
Twins of rayless graphs, Journal of Combinatorial Theory, Series B
  • Volume 101,
  • 2011
The structure of rayless graphs
The random graph, Algorithms and Combinatorics
  • Volume 14,
  • 1997