A total dominating set of a graph G = (V, E) is a subset D of V such that every vertex in V is adjacent to at least one vertex in D. The total domination number of G, denoted by γ t (G), is the minimum cardinality of a total dominating set of G. A near-triangulation is a biconnected planar graph that admits a plane embedding such that all of its faces are triangles except possibly the outer face. We show in this paper that γ t (G) ≤ 2n 5 for any near-triangulation G of order n ≥ 5, with two exceptions.

Abstract. Stabbing a set S of n segments in the plane by a line is a well-known 8 problem. In this paper we consider the variation where the stabbing object is a cir-9 cle instead of a line. We show that the problem is tightly connected to two cluster 10Voronoi diagrams, in particular, the Hausdorff and the farthest-color Voronoi dia-11 gram. Based on these diagrams, we provide a method to compute a representation 12 of all the combinatorially different stabbing circles for S, and the stabbing circles 13 with maximum and minimum radius. We give conditions under which our method 14 is fast. These conditions are satisfied if the segments in S are parallel, resulting in 15 a O(n log 2 n) time and O(n) space algorithm. We also observe that the stabbing

Abstract. Stabbing a set S of n segments in the plane by a line is a well-known 8 problem. In this paper we consider the variation where the stabbing object is a cir-9 cle instead of a line. We show that the problem is tightly connected to two cluster

We undertake a study on computing Hamiltonian alternating cycles and paths on bicolored point sets. This has been an intensively studied problem, not always with a solution, when the paths and cycles are also required to be plane. In this paper, we relax the constraint on the cycles and paths from being plane to being 1-plane, and deal with the same type of questions as those for the plane case, obtaining a remarkable variety of results. For point sets in general position, our main result is that it is always possible to obtain a 1-plane Hamiltonian alternating cycle. When the point set is in convex position, we prove that every Hamiltonian alternating cycle with minimum number of crossings is 1-plane, and provide O(n) and O(n2) time algorithms for computing, respectively, Hamiltonian alternating cycles and paths with minimum number of crossings.Peer ReviewedPostprint (author's final draft

In this paper we study the metric dimension problem in maximal outerplanar graphs. Concretely, if β(G) is the metric dimension of a maximal outerplanar graph G of order n, we prove that 2 ≤ β(G) ≤ 2n 5 and that the bounds are tight. We also provide linear algorithms to decide whether the metric dimension of G is 2 and to build a resolving set S of size 2n 5 for G. Moreover, we characterize the maximal outerplanar graphs with metric dimension 2.