AT-free graphs: linear bounds for the oriented diameter

  • Authors:

  • Fedor V. Fomin;Martín Matamala;Erich Prisner;Ivan Rapaport

  • Affiliations:

  • Centro de Modelamiento Matemático, Universidad de Chile and UMR 2071-CNRS, Casilla 170-3, Correo 3, Santiago, Chile;Centro de Modelamiento Matematico, Univ. de Chile and UMR 2071-CNRS, Santiago, Chile and Dept. de Ingenieria Matematica, Facultad de Ciencias Fisicas y Matematicas, Univ. de Chile, Santiago, Chile;Department of Mathematics, University of Louisville, Louisville, KY;Centro de Modelamiento Matematico, Univ. de Chile and UMR 2071-CNRS, Santiago, Chile and Dept. de Ingenieria Matematica, Facultad de Ciencias Fisicas y Matematicas, Univ. de Chile, Santiago, Chile

  • Venue:
  • Discrete Applied Mathematics - Brazilian symposium on graphs, algorithms and combinatorics
  • Year:
  • 2004

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Abstract

Let G be a bridgeless connected undirected (b.c.u.) graph. The oriented diameter of G, OD(G), is given by OD(G)=min{diam(H): H is an orientation of G}, where diam(H) is the maximum length computed over the lengths of all the shortest directed paths in H. This work starts with a result stating that, for every b.c.u, graph G, its oriented diameter OD(G) and its domination number γ(G) are linearly related as follows: OD(G)≤ 9γ(G)-5.Since-as shown by Corneil et al. (SIAM J. Discrete Math. 10 (1997) 399)-γ(G)≤ diam(G) for every AT-free graph G, it follows that OD(G) ≤ 9 diam(G)-5 for every b.c.u. AT-free graph G. Our main result is the improvement of the previous linear upper bound. We show that OD(G) ≤ 2 diam(G)+11 for every b.c.u. AT-free graph G. For some subclasses we obtain better bounds: OD(G) ≤ 3/2 diam(G)+25/2 for every interval b.c.u. graph G, and OD(G) ≤ 5/4 diam(G)+ 29/2 for every 2-connected interval b.c.u. graph G. We prove that, for the class of b.c.u. AT-free graphs and its previously mentioned subclasses, all our bounds are optimal (up to additive constants).