From Path Graphs to Directed Path Graphs

• Steven Chaplick
• Marisa Gutierrez
• Benjamin Lévêque
• Silvia B. Tondato
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6410)

Abstract

We present a linear time algorithm to greedily orient the edges of a path graph model to obtain a directed path graph model (when possible). Moreover we extend this algorithm to find an odd sun when the method fails. This algorithm has several interesting consequences concerning the relationship between path graphs and directed path graphs. One is that for a directed path graph, path graph models and directed path graph models are the same. Another consequence concerns the difference between path graphs and directed path graphs in terms of forbidden induced subgraphs. This can be used to deduce the forbidden induced subgraph characterization of directed path graphs from the forbidden induced subgraph characterization of path graphs. The last consequence is algorithmic and shows that the recognition of directed path graphs is not more difficult than the recognition of path graphs.

Keywords

Directed Path Maximal Clique Intersection Graph Interval Graph Chordal Graph
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

1. 1.
Chaplick, S.: PQR-trees and undirected path graphs, M.Sc. Thesis, Dept. of Computer Science, University of Toronto, Canada (2008)Google Scholar
2. 2.
Dahlhaus, E., Bailey, G.: Recognition of path graphs in linear time. In: 5th Italian Conference on Theoretical Computer Science (Revello, 1995), pp. 201–210. World Sci., Publishing, River Edge (1996)Google Scholar
3. 3.
Gavril, F.: The intersection graphs of subtrees in trees are exactly the chordal graphs. J. Combin. Theory B 16, 47–56 (1974)
4. 4.
Gavril, F.: A recognition algorithm for the intersection graphs of paths in trees. Discrete Math. 23, 211–227 (1978)
5. 5.
Golumbic, M.C.: Algorithmic graph theory and perfect graphs. Annals Disc. Math. 57 (2004)Google Scholar
6. 6.
Fulkerson, D.R., Gross, O.A.: Incidence matrices and interval graphs. Pacific J. Math. 15, 835–855 (1965)
7. 7.
Lévêque, B., Maffray, F., Preissmann, M.: Characterizing path graphs by forbidden induced subgraphs. Journal of Graph Theory 62, 369–384 (2009)
8. 8.
McKee, T.A., McMorris, F.R.: Topics in intersection graph theory. SIAM Monographs on Discrete Mathematics and Applications, Philadelphia (1999)Google Scholar
9. 9.
Monma, C.L., Wei, V.K.: Intersection graphs of paths in a tree. Journal of Combinatorial Theory B 41, 141–181 (1986)
10. 10.
Panda, B.S.: The forbidden subgraph characterization of directed vertex graphs. Discrete Mathematics 196, 239–256 (1999)
11. 11.
Schäffer, A.A.: A faster algorithm to recognize undirected path graphs. Discrete Appl. Math. 43, 261–295 (1993)
12. 12.
Tondato, S.B.: Grafos Cordales: Árboles clique y Representaciones canónicas, Doctoral Thesis, Universidad Nacional de La Plata, Argentina (2009) (in spanish) Google Scholar

Authors and Affiliations

• Steven Chaplick
• 1
• Marisa Gutierrez
• 2
• Benjamin Lévêque
• 3
• Silvia B. Tondato
• 4