Localization Game on Geometric and Planar Graphs

The main topic of this paper is motivated by a localization problem in cellular networks. Given a graph G we want to localize a walking agent by checking his distance to as few vertices as possible. The model we introduce is based on a pursuit graph game that resembles the famous Cops and Robbers game. It can be considered as a game theoretic variant of the metric dimension of a graph. We provide upper bounds on the related graph invariant ζ(G), defined as the least number of cops needed to localize the robber on a graph G, for several classes of graphs (trees, bipartite graphs, etc). Our main result is that, surprisingly, there exists planar graphs of treewidth 2 and unbounded ζ(G). On a positive side, we prove that ζ(G) is bounded by the pathwidth of G. We then show that the algorithmic problem of determining ζ(G) is NP-hard in graphs with diameter at most 2. Finally, we show that at most one cop can approximate (arbitrary close) the location of the robber in the Euclidean plane

Let us play the cleaning game

Non UBCUnreviewedAuthor affiliation: Technical University of LodzFacult

Small on-line Ramsey numbers---a new approach

In this note, we revisit the problem of calculating small on-line Ramsey numbers R(G,H). A new approach is proposed that reduces the running time of the algorithm determining that R(K_3,K_4)=17 by a factor of at least 2*10^6 comparing to the previously used approach. Using high performance computing networks, we determined that R(K_4,K_4) <= 26, R(K_3,K_5) < 25, and that R(K_3,K_3,K_3) <= 20 for a natural generalization to three colours. All graphs on 3 or 4 vertices are investigated as well, including non-symmetric cases