Centroidal localization game

One important problem in a network is to locate an (invisible) moving entity by using distance-detectors placed at strategical locations. For instance, the metric dimension of a graph $G$ is the minimum number $k$ of detectors placed in some vertices $\{v_1,\cdots,v_k\}$ such that the vector $(d_1,\cdots,d_k)$ of the distances $d(v_i,r)$ between the detectors and the entity's location $r$ allows to uniquely determine $r \in V(G)$. In a more realistic setting, instead of getting the exact distance information, given devices placed in $\{v_1,\cdots,v_k\}$, we get only relative distances between the entity's location $r$ and the devices (for every $1\leq i,j\leq k$, it is provided whether $d(v_i,r) >$, $<$, or $=$ to $d(v_j,r)$). The centroidal dimension of a graph $G$ is the minimum number of devices required to locate the entity in this setting. We consider the natural generalization of the latter problem, where vertices may be probed sequentially until the moving entity is located. At every turn, a set $\{v_1,\cdots,v_k\}$ of vertices is probed and then the relative distances between the vertices $v_i$ and the current location $r$ of the entity are given. If not located, the moving entity may move along one edge. Let $\zeta^* (G)$ be the minimum $k$ such that the entity is eventually located, whatever it does, in the graph $G$. We prove that $\zeta^* (T)\leq 2$ for every tree $T$ and give an upper bound on $\zeta^*(G\square H)$ in cartesian product of graphs $G$ and $H$. Our main result is that $\zeta^* (G)\leq 3$ for any outerplanar graph $G$. We then prove that $\zeta^* (G)$ is bounded by the pathwidth of $G$ plus 1 and that the optimization problem of determining $\zeta^* (G)$ is NP-hard in general graphs. Finally, we show that approximating (up to any constant distance) the entity's location in the Euclidean plane requires at most two vertices per turn

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 \emph{metric dimension} of a graph. We provide upper bounds on the related graph invariant $\zeta (G)$, 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 $\zeta (G)$. On a positive side, we prove that $\zeta (G)$ is bounded by the pathwidth of $G$. We then show that the algorithmic problem of determining $\zeta (G)$ 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

The first player wins the one-colour triangle avoidance game on 16 vertices

We consider the one-colour triangle avoidance game. Using a high performance computing network, we showed that the first player can win the game on 16 vertices

Bounds and constructions for n-e.c. tournaments

Few families of tournaments satisfying the $n$-e.c.\ adjacency property are known. We supply a new random construction for generating infinite families of vertex-transitive $n$-e.c.\ tournaments by considering circulant tournaments. Switching is used to generate exponentially many $n$-e.c.\ tournaments of certain orders. With aid of a computer search, we demonstrate that there is a unique minimum order $3$-e.c.\ tournament of order $19,$ and there are no $3$-e.c.\ tournaments of orders $20,$ $21,$ and $22.

On the Solvability of the Discrete <i>s</i>(<i>x</i>,·)-Laplacian Problems on Simple, Connected, Undirected, Weighted, and Finite Graphs

We use the finite dimensional monotonicity methods in order to investigate problems connected with the discrete sx,·-Laplacian on simple, connected, undirected, weighted, and finite graphs with nonlinearities given in a non-potential form. Positive solutions are also considered

Bounds and constructions for n-e.c. tournaments

Few families of tournaments satisfying the $n$-e.c.\ adjacency property are known. We supply a new random construction for generating infinite families of vertex-transitive $n$-e.c.\ tournaments by considering circulant tournaments. Switching is used to generate exponentially many $n$-e.c.\ tournaments of certain orders. With aid of a computer search, we demonstrate that there is a unique minimum order $3$-e.c.\ tournament of order $19,$ and there are no $3$-e.c.\ tournaments of orders $20,$ $21,$ and $22.