AT-free graphs: linear bounds for the oriented diameter

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 (G) are linearly related as follows: OD(G)6 9 (G)− 5. Since—as shown by Corneil et al. (SIAM J. Discrete Math. 10 (1997) 399)— (G)6diam(G) for every AT-free graph G, it follows that OD(G)6 9diam(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)6 2diam(G)+11 for every b.c.u. AT-free graph G. For some subclasses we obtain better bounds: OD(G)6 2 diam(G)+ 25 2 for every interval b.c.u. graph G, and OD(G)6 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). ? 2003 Elsevier B.V. All rights reserved. MSC: 05C12; 05C20; 05C69; 05C62