Isomorphism problem for geometric intersection graphs

Bc. Deniz Agaoglu, B.Sc.

Master's thesis

Abstract:

Graph Isomorphism Problem is one of the well-known problems in graph theory. Although it is not known whether it belongs to P or not, it has been shown to be solvable in polynomial and even in linear time for various graph classes. In particular, the isomorphism problem is solvable in polynomial-time for interval graphs and it is as hard as on all graphs for chordal graphs (i.e., GI-complete). Our main motivation is to identify the complexity of the isomorphism problem for geometric graph classes between these two classes. In this thesis, we focus on S_d- and T-graphs which are subclasses of H-graphs, introduced by Biro, Hujter and Tuza \cite{biro}, and prove that the S_d-graphs of bounded clique size can be solved in FPT-time, and for T-graphs of bounded clique size it is solvable in FPT-time when the number of leaves of T are bounded. The same tractability questions for S_d- and T-graphs of unbounded clique size, but bounded number of leaves are related to the unsolved problem of isomorphism of posets of bounded width. If the isomorphism for S_d-graphs can be solved in FPT-time, then it can be used to solve the isomorphism problem for posets of width d. Then, we prove that the isomorphism problem for posets of width 2 can be solved in polynomial time and show that it is not directly extendible to posets of higher widths. Moreover, we prove that the isomorphism problem for CARC-graphs of bounded degree and without a dominating triangle or edge can be solved in FPT-time.

Keywords

interval graphs, chordal graphs, geometric graphs, intersection graphs, isomorphism

Language used: English

Date on which the thesis was submitted / produced: 21. 5. 2018

Thesis defence

• Date of defence: 22. 6. 2018
• Supervisor: prof. RNDr. Petr Hliněný, Ph.D.
• Reader: Dr. rer. nat. Achim Blumensath

Masaryk University

Faculty of Informatics

Master programme / field:
Informatics / Information Technology Security (eng.)