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A dominating set of a graph G is a subset D of vertices such that every vertex not in D is adjacent to at least one vertex in D. A dominating set D is paired if the subgraph induced by its vertices has a perfect matching, and semipaired if every vertex in D is paired with exactly one other vertex in D that is within distance 2 from it. The paired domination number, denoted by ¿pr(G), is the minimum cardinality of a paired dominating set of G, and the semipaired domination number, denoted by ¿pr2(G), is the minimum cardinality of a semipaired 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 ¿pr(G) = 2b n 4 c for any neartriangulation G of order n = 4, and that with some exceptions, ¿pr2(G) = b 2n 5 c for any near-triangulation G of order n = 5.

In this work we study the metric dimension and the location-domination number of maximal outerplanar graphs. Concretely, we determine tight upper and lower bounds on the metric dimension and characterize those maximal outerplanar graphs attaining the lower bound. We also give a lower bound on the location-domination number of maximal outerplanar graphs.

We give explicit formulae for obtaining the binary sequences which produce Steinhaus triangles and generalized Pascal triangles with rotational and dihedral symmetries.

In this thesis, we study Steinhaus's problem. We begin with the definitions of Steinhaus triangle and balanced binary sequence, and we study sequences with different properties. We then show three different proofs of the existence of balanced binary sequences of any length n ≡ 0 or 3 (mod 4). In the last chapter, we present Molluzzo s problem, a generalization of Steinhaus s problem. We study arithmetic progressions and antisymmetric sequences and their associated generalized triangles. On the one hand, we show the existence of balanced sequences in Z/3 k Z of any valid length. On the other hand, we see different cases in which Molluzzo s problem is answered negatively.