Diameter Determination on Restricted Graph Families
Determining the diameter of a graph is a fundamental graph operation, yet no efficient (i.e. quadratic time) algorithm is known. In this paper, we examine the diameter problem on chordal and AT-free graphs and show that a very simple (linear time) 2-sweep Lex-BFS algorithm identifies a vertex of maximum eccentricity unless the given graph has a specified induced subgraph (it was previously known that a single Lex-BFS algorithm is guaranteed to end at a vertex that is within 1 of the diameter for chordal and AT-free graphs). As a consequence of the forbidden induced subgraph result on chordal graphs, our algorithm is guaranteed to work optimally for directed path graphs (it was previously known that a single LexBFS algorithm is guaranteed to work optimally for interval graphs).
KeywordsIntersection Graph Switching Point Interval Graph Chordal Graph Arbitrary Graph
Unable to display preview. Download preview PDF.
- 2.Chepoi, V.D., Dragan, F.F.: Disjoint set problem (1992) (unpublished)Google Scholar
- 3.Corneil, D.G., Olariu, S., Stewart, L.: Linear time algorithms for dominating pairs in asteroidal triple-free graphs. Technical Report 294-95, University of Toronto, To appear in SIAM Journal of Computing (January 1995)Google Scholar
- 5.Dragan, F., Nicolai, F., Brandstädt, A.: Lex-BFS orderings and powers of graphs. In: D’Amore, F., Marchetti-Spaccamela, A., Franciosa, P.G. (eds.) WG 1996. LNCS, vol. 1197, pp. 166–180. Springer, Heidelberg (1997)Google Scholar
- 6.Dragan, F.: Almost diameter in hhd-free graphs in linear time via lex-bfs. In: Optimal Discrete Structures and Algorithms. University of Rostock, Germany (1997)Google Scholar
- 11.Lekkerkerker, C.G., Boland, J.C.: Representation of a finite graph by a set of intervals on the real line. Fund. Math. 51, 45–64 (1962)Google Scholar
- 12.Lesk, M.: Couplages maximaux et diamètre de graphes. PhD thesis, Université Pierre et Marie Curie, Paris 6 (October 1984)Google Scholar
- 13.Paul, C.: Parcours en largeur lexicographique: un algorithme de partitionnement, application aux graphes et généralisation. PhD thesis, LIRMM, Université de Montpellier II (1998)Google Scholar