A note on the convexity number for complementary prisms

In the geodetic convexity, a set of vertices $S$ of a graph $G$ is $\textit{convex}$ if all vertices belonging to any shortest path between two vertices of $S$ lie in $S$. The cardinality $con(G)$ of a maximum proper convex set $S$ of $G$ is the $\textit{convexity number}$ of $G$. The $\textit{complementary prism}$ $G\overline{G}$ of a graph $G$ arises from the disjoint union of the graph $G$ and $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. In this work, we we prove that the decision problem related to the convexity number is NP-complete even restricted to complementary prisms, we determine $con(G\overline{G})$ when $G$ is disconnected or $G$ is a cograph, and we present a lower bound when $diam(G) \neq 3$.Comment: 10 pages, 2 figure

On the Graceful Game

A graceful labeling of a graph $G$ with $m$ edges consists of labeling the vertices of $G$ with distinct integers from $0$ to $m$ such that, when each edge is assigned as induced label the absolute difference of the labels of its endpoints, all induced edge labels are distinct. Rosa established two well known conjectures: all trees are graceful (1966) and all triangular cacti are graceful (1988). In order to contribute to both conjectures we study graceful labelings in the context of graph games. The Graceful game was introduced by Tuza in 2017 as a two-players game on a connected graph in which the players Alice and Bob take turns labeling the vertices with distinct integers from 0 to $m$. Alice's goal is to gracefully label the graph as Bob's goal is to prevent it from happening. In this work, we study winning strategies for Alice and Bob in complete graphs, paths, cycles, complete bipartite graphs, caterpillars, prisms, wheels, helms, webs, gear graphs, hypercubes and some powers of paths

Locating-Domination in Complementary Prisms.

Let G = (V (G), E(G)) be a graph and G̅ be the complement of G. The complementary prism of G, denoted GG̅, is the graph formed from the disjoint union of G and G̅ by adding the edges of a perfect matching between the corresponding vertices of G and G̅. A set D ⊆ V (G) is a locating-dominating set of G if for every u ∈ V (G)D, its neighborhood N(u)⋂D is nonempty and distinct from N(v)⋂D for all v ∈ V (G)D where v ≠ u. The locating-domination number of G is the minimum cardinality of a locating-dominating set of G. In this thesis, we study the locating-domination number of complementary prisms. We determine the locating-domination number of GG̅ for specific graphs and characterize the complementary prisms with small locating-domination numbers. We also present bounds on the locating-domination numbers of complementary prisms

On the coarse classification of tight contact structures

We present a sketch of the proof of the following theorems: (1) Every 3-manifold has only finitely many homotopy classes of 2-plane fields which carry tight contact structures. (2) Every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures.Comment: 12 pages, to appear in the 2001 Georgia International Topology Conference proceeding

A laser velocimeter system for large-scale aerodynamic testing

A unique laser velocimeter was developed that is capable of sensing two orthogonal velocity components from a variable remote distance of 2.6 to 10 m for use in the 40- by 80-Foot and 80- by 120-Foot Wind Tunnels and the Outdoor Aerodynamic Research Facility at Ames Research Center. The system hardware, positioning instrumentation, and data acquisition equipment are described in detail; system capabilities and limitations are discussed; and expressions for systematic and statistical accuracy are developed. Direct and coupled laboratory measurements taken with the system are compared with measurements taken with a laser velocimeter of higher spatial resolution, and sample data taken in the open circuit exhaust flow of a 1/50-scale model of the 80- by 120-Foot Wind Tunnel are presented

Italian Domination in Complementary Prisms

Let $G$ be any graph and let $\overline{G}$ be its complement. The complementary prism of $G$ is formed from the disjoint union of a graph $G$ and its complement $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. An Italian dominating function on a graph $G$ is a function such that $f \, : \, V \to \{ 0,1,2 \}$ and for each vertex $v \in V$ for which $f(v)=0$, it holds that $\sum_{u \in N(v)} f(u) \geq 2$. The weight of an Italian dominating function is the value $f(V)=\sum_{u \in V(G)}f(u)$. The minimum weight of all such functions on $G$ is called the Italian domination number. In this thesis we will study Italian domination in complementary prisms. First we will present an error found in one of the references. Then we will define the small values of the Italian domination in complementary prisms, find the value of the Italian domination number in specific families of graphs complementary prisms, and conclude with future problems