We discuss several results concerning on-line algorithms for ordered sets and comparability graphs. The main new result is on the problem of on-line transitive orientation. We view on-line transitive orientation of a comparability graph G as a two-person game. In the ith round of play, 1 ≤ i ≤ | V(G)|, player A names a graph Gi such that Gi is isomorphic to a subgraph of G, |V(Gi)| = i, and Gi-1 is an induced subgraph of Gi if i > 1. Player B must respond with a transitive orientation of Gi which extends the transitive orientation given to Gi-1 in the previous round of play. Player A wins if and only if player B fails to give a transitive orientation to Gi for some i, 1 ≤ i ≤ |V(G)|. Our main result shows that player A has at most three winning moves. We also discuss applications to on-line clique covering of comparability graphs, and we mention some open problems. © 1995.
Penrice, S. G. (1995). On-line algorithms for ordered sets and comparability graphs. Discrete Applied Mathematics, 60(1–3), 319–329. https://doi.org/10.1016/0166-218X(94)00062-I