Abstract

In this paper, we introduce the class of chordal probe graphs which are a generalization of both interval probe graphs and chordal graphs. A graph G is chordal probe if its vertices can be partitioned into two sets P (probes) and N (non-probes) where N is a stable set and such that G can be extended to a chordal graph by adding edges between non-probes. We show that chordal probe graphs may contain neither an odd-length chordless cycle nor the complement of a chordless cycle, hence they are perfect graphs. We present a complete hierarchy with separating examples for chordal probe and related classes of graphs. We give polynomial time recognition algorithms for the subfamily of chordal probe graphs which are also weakly chordal, first in the case of a fixed given partition of the vertices into probes and non-probes, and second in the more general case where no partition is given.

Author Keywords: Chordal graphs; Probe graphs; Interval graphs

Article Outline

1. Introduction

1.1. Basic definitions

1.2. Chordal probe graphs: definitions and examples

1.3. Outline of the paper

2. First results

3. Even-chordal graphs

4. Hierarchy of chordal probe and even-chordal graphs

5. Recognition of those chordal probe graphs which are also even-chordal with respect to a given partition

6. The C₄-connectivity relation

7. Recognition of those chordal probe graphs which are also even-chordal without being given the partition

8. Conclusion

References