Interval Graphs with Containment Restrictions
Abstract
An interval graph is proper iff it has a representation in which no interval contains another. Fred Roberts characterized the proper interval graphs as those containing no induced star $K_{1,3}$. Proskurowski and Telle have studied $q$-proper graphs, which are interval graphs having a representation in which no interval is properly contained in more than $q$ other intervals. Like Roberts they found that their classes of graphs where characterized, each by a single minimal forbidden subgraph. This paper initiates the study of $p$-improper interval graphs where no interval contains more than $p$ other intervals. This paper will focus on a special case of $p$-improper interval graphs for which the minimal forbidden subgraphs are readily described. Even in this case, it is apparent that a very wide variety of minimal forbidden subgraphs are possible.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2011
- DOI:
- 10.48550/arXiv.1109.6675
- arXiv:
- arXiv:1109.6675
- Bibcode:
- 2011arXiv1109.6675B
- Keywords:
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- Mathematics - Combinatorics;
- 05C75
- E-Print:
- 12 pages