Intrinsically knotted graphs and connected domination

TL;DR

This paper classifies certain linklessly embeddable graphs, explores their complements' intrinsic knotting properties, and investigates connected domination numbers, providing new insights into graph embedding and topological properties.

Contribution

It classifies maximal linklessly embeddable graphs of order 12, analyzes their complements' intrinsic knotting, and addresses open questions on minimal order of 3-non-compliant graphs.

Findings

Maximal linklessly embeddable graphs of order 12 are classified.

Complements of these graphs are intrinsically knotted.

Complements of knotlessly embeddable graphs of order ≥15 are intrinsically knotted.

Abstract

We classify all the maximal linklessly embeddable graphs of order 12 and show that their complements are all intrinsically knotted. We derive results about the connected domination numbers of a graph and its complement. We provide an answer to an open question about the minimal order of a 3-non-compliant graph. We prove that the complements of knotlessly embeddable graphs of order at least 15 are all intrinsically knotted. We provide results on general $k$-non-compliant graphs and leave a set of open questions for further exploration of the subject.