Paired and semipaired domination in triangulations

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 $\gamma_{pr}(G)$, is the minimum cardinality of a paired dominating set of $G$, and the semipaired domination number, denoted by $\gamma_{pr2}(G)$, 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 $\gamma_{pr}(G) \le 2 \lfloor \frac{n}{4} \rfloor$ for any near-triangulation $G$ of order $n\ge 4$, and that with some exceptions, $\gamma_{pr2}(G) \le \lfloor \frac{2n}{5} \rfloor$ for any near-triangulation $G$ of order $n\ge 5$

Terminal Phosphanido Rhodium Complexes Mediating Catalytic P—P and P—C Bond Formation

Complexes with terminal phosphanido (M-PR2) functionalities are believed to be crucial intermediates in new catalytic processes involving the formation of P-P and P-C bonds. We showcase here the isolation and characterization of mononuclear phosphanide rhodium complexes ([RhTp(H)(PR2)L]) that result from the oxidative addition of secondary phosphanes, a reaction that was also explored computationally. These compounds are active catalysts for the dehydrocoupling of PHPh2 to Ph2P-PPh2. The hydrophosphination of dimethyl maleate and the unactivated olefin ethylene is also reported. Reliable evidence for the prominent role of mononuclear phosphanido rhodium species in these reactions is also provided.The generous financial support from MICINN/FEDER (Project CTQ2011-22516), Gobierno de Aragón/FSE (GA/FSE, Inorganic Molecular Architecture Group, E70), and NWO-CW (VICI project 016.122.613; BdB) is gratefully acknowledged. A.M.G. and A.L.S. thank Gobierno de Aragón and MEC, respectively, for fellowships.Peer reviewe

Rhodium complexes in p-C bond formation: Key role of a hydrido ligand

Olefin hydrophosphanation is an attractive route for the atom-economical synthesis of functionalized phosphanes. This reaction involves the formation of P-C and H-C bonds. Thus, complexes that contain both hydrido and phosphanido functionalities are of great interest for the development of effective and fast catalysts. Herein, we showcase the excellent activity of one of them, [Rh(Tp)H(PMe3)(PPh2)] (1), in the hydrophosphanation of a wide range of olefins. In addition to the required nucleophilicity of the phosphanido moiety to accomplish the P-C bond formation, the key role of the hydride ligand in 1 has been disclosed by both experimental results and DFT calculations. An additional Rh-H··· C stabilization in some intermediates or transition states favors the hydrogen transfer reaction from rhodium to carbon to form the H-C bond. Further support for our proposal arises from the poor activity exhibited by the related chloride complex [Rh(Tp)Cl(PMe3)(PPh2)] as well as from stoichiometric and kinetic studies

Metric Dimension of Maximal Outerplanar Graphs

In this paper, we study the metric dimension problem in maximal outerplanar graphs. Concretely, if β(G) denotes the metric dimension of a maximal outerplanar graph G of order n, we prove that 2≤β(G)≤⌈2n5⌉ 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 ⌈2n5⌉ for G. Moreover, we characterize all maximal outerplanar graphs with metric dimension 2

Maximum rectilinear convex subsets

Let P be a set of n points in the plane. We consider a variation of the classical Erd\H os-Szekeres problem, presenting efficient algorithms with O(n3) running time and O(n2) space complexity that compute (1) a subset S of P such that the boundary of the rectilinear convex hull of S has the maximum number of points from P, (2) a subset S of P such that the boundary of the rectilinear convex hull of S has the maximum number of points from P and its interior contains no element of P, (3) a subset S of P such that the rectilinear convex hull of S has maximum area and its interior contains no element of P, and (4) when each point of P is assigned a weight, positive or negative, a subset S of P that maximizes the total weight of the points in the rectilinear convex hull of S. We also revisit the problems of computing a maximum area orthoconvex polygon and computing a maximum area staircase polygon, amidst a point set in a rectangular domain. We obtain new and simpler algorithms to solve both problems with the same complexity as in the state of the art

K1,3-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 K1, 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 ¿ B can be K1, 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 ¿ B, like {1, 3}-equitable sets (i.e., r = 3b or b = 3r) and linearly separable sets, that can be K1, 3-covered. (2) If r = 3g + h, b = 3h + g and the points in R ¿ B are in convex position, then at least r + b - 4 points can be K1, 3-covered, and this bound is tight. (3) There are arbitrarily large point sets R ¿ B in general position, with r = b + 1, such that at most r + b - 5 points can be K1, 3-covered. (4) If b = r = 3b, then at least 9 8 (r + b- 8) points of R ¿ B can be K1, 3-covered. For r > 3b, there are too many red points and at least r - 3b of them will remain uncovered in any K1, 3-covering. Furthermore, in all the cases we provide efficient algorithms to compute the corresponding coverings

Maximum rectilinear convex subsets

Let P be a set of n points in the plane. We consider a variation of the classical Erdos-Szekeres problem, presenting efficient algorithms with (formula presented) running time and (formula presented) space complexity that compute: (1) A subset S of P such that the boundary of the rectilinear convex hull of S has the maximum number of points from P, (2) a subset S of P such that the boundary of the rectilinear convex hull of S has the maximum number of points from P and its interior contains no element of P, (3) a subset S of P such that the rectilinear convex hull of S has maximum area and its interior contains no element of P, and (4) when each point of P is assigned a weight, positive or negative, a subset S of P that maximizes the total weight of the points in the rectilinear convex hull of S