Automorphism Groups of Comparability Graphs

TL;DR

This paper characterizes the automorphism groups of permutation graphs, explores their relation to comparability graphs, and demonstrates that for dimensions four and above, any finite group can be realized as an automorphism group, with isomorphism testing being GI-complete.

Contribution

It provides a characterization of automorphism groups of permutation graphs and shows the universality of automorphism groups for graphs with dimension at least 4.

Findings

Automorphism groups of permutation graphs are characterized similarly to trees, with additional groups due to extra operations.

For dimensions ≥ 4, any finite group can be realized as an automorphism group of some graph.

Graph isomorphism testing for k-DIM graphs is GI-complete.

Abstract

Comparability graphs are graphs which have transitive orientations. The dimension of a poset is the least number of linear orders whose intersection gives this poset. The dimension ${\rm dim}(X)$ of a comparability graph $X$ is the dimension of any transitive orientation of X, and by $k$-DIM we denote the class of comparability graphs $X$ with ${\rm dim}(X) \le k$. It is known that the complements of comparability graphs are exactly function graphs and permutation graphs equal 2-DIM. In this paper, we characterize the automorphism groups of permutation graphs similarly to Jordan's characterization for trees (1869). For permutation graphs, there is an extra operation, so there are some extra groups not realized by trees. For $k \ge 4$, we show that every finite group can be realized as the automorphism group of some graph in $k$-DIM, and testing graph isomorphism for $k$-DIM is GI-complete.