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

K1,3K_{1,3}-covering red and blue points in the plane

We say that a finite set of red and blue points in the plane in general position can be $K_{1,3}$-covered if the set can be partitioned into subsets of size $4$, with $3$ points of one color and $1$ point of the other color, in such a way that, if at each subset the fourth point is connected by straight-line segments to the same-colored points, then the resulting set of all segments has no crossings. We consider the following problem: Given a set $R$ of $r$ red points and a set $B$ of $b$ blue points in the plane in general position, how many points of $R\cup B$ can be $K_{1,3}$-covered? and we prove the following results: (1) If $r=3g+h$ and $b=3h+g$, for some non-negative integers $g$ and $h$, then there are point sets $R\cup B$, like $\{1,3\}$-equitable sets (i.e., $r=3b$ or $b=3r$) and linearly separable sets, that can be $K_{1,3}$-covered. (2) If $r=3g+h$, $b=3h+g$ and the points in $R\cup B$ are in convex position, then at least $r+b-4$ points can be $K_{1,3}$-covered, and this bound is tight. (3) There are arbitrarily large point sets $R\cup B$ in general position, with $r=b+1$, such that at most $r+b-5$ points can be $K_{1,3}$-covered. (4) If $b\le r\le 3b$, then at least $\frac{8}{9}(r+b-8)$ points of $R\cup B$ can be $K_{1,3}$-covered. For $r>3b$, there are too many red points and at least $r-3b$ of them will remain uncovered in any $K_{1,3}$-covering. Furthermore, in all the cases we provide efficient algorithms to compute the corresponding coverings.Comment: 29 pages, 10 figures, 1 tabl

The alternating path problem revisited

It is well known that, given n red points and n blue points on a circle, it is not always possible to find a plane geometric Hamiltonian alternating path. In this work we prove that if we relax the constraint on the path from being plane to being 1-plane, then the problem always has a solution, and even a Hamiltonian alternating cycle can be obtained on all instances. We also extend this kind of result to other configurations and provide remarks on similar problems.Ministerio de Economía y CompetitividadGeneralitat de CatalunyaEuropean Science FoundationMinisterio de Ciencia e InnovaciónJunta de Andalucía (Consejería de Innovación, Ciencia y Empresa

On the expected number of perfect matchings in cubic planar graphs

A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that a bridgeless cubic graph has exponentially many perfect matchings. It was solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the special case of cubic planar graphs. In our work we consider random bridgeless cubic planar graphs with the uniform distribution on graphs with $n$ vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically $c\gamma^n$, where $c>0$ and $\gamma \sim 1.14196$ is an explicit algebraic number. We also compute the expected number of perfect matchings in (non necessarily bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs. Our starting point is a correspondence between counting perfect matchings in rooted cubic planar maps and the partition function of the Ising model in rooted triangulations.Comment: 19 pages, 4 figure

Monochromatic geometric k-factors for bicolored point sets with auxiliary points

Given a bicolored point set S, it is not always possible to construct a monochromatic geometric planar k-factor of S. We consider the problem of finding such a k-factor of S by using auxiliary points. Two types are considered: white points whose position is fixed, and Steiner points which have no fixed position. Our approach provides algorithms for constructing those k-factors, and gives bounds on the number of auxiliary points needed to draw a monochromatic geometric planar k-factor of S

Quasi-Parallel Segments and Characterization of Unique Bichromatic Matchings

Given n red and n blue points in general position in the plane, it is well-known that there is a perfect matching formed by non-crossing line segments. We characterize the bichromatic point sets which admit exactly one non-crossing matching. We give several geometric descriptions of such sets, and find an O(nlogn) algorithm that checks whether a given bichromatic set has this property.Comment: 31 pages, 24 figure

Gap-ETH-Tight Approximation Schemes for Red-Green-Blue Separation and Bicolored Noncrossing Euclidean Travelling Salesman Tours

In this paper, we study problems of connecting classes of points via noncrossing structures. Given a set of colored terminal points, we want to find a graph for each color that connects all terminals of its color with the restriction that no two graphs cross each other. We consider these problems both on the Euclidean plane and in planar graphs. On the algorithmic side, we give a Gap-ETH-tight EPTAS for the two-colored traveling salesman problem as well as for the red-blue-green separation problem (in which we want to separate terminals of three colors with two noncrossing polygons of minimum length), both on the Euclidean plane. This improves the work of Arora and Chang (ICALP 2003) who gave a slower PTAS for the simpler red-blue separation problem. For the case of unweighted plane graphs, we also show a PTAS for the two-colored traveling salesman problem. All these results are based on our new patching procedure that might be of independent interest. On the negative side, we show that the problem of connecting terminal pairs with noncrossing paths is NP-hard on the Euclidean plane, and that the problem of finding two noncrossing spanning trees is NP-hard in plane graphs.Comment: 36 pages, 15 figures (colored