Seymour's second neighborhood conjecture for tournaments missing a generalized star

Seymour's Second Neighborhood Conjecture asserts that every digraph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. We prove its weighted version for tournaments missing a generalized star. As a consequence the weighted version holds for tournaments missing a sun, star, or a complete graph.Comment: Accepted for publication in Journal of Graph Theory in 24 June 201

An algorithm proposal for a minimum cost SDR multi-standard system using graph theory

International audienceThe design of future multi-standard systems remains a challenge due to increasing flexibility requirements. Promising solutions include designing flexible radio architectures that exploit common aspects between the different set of standards cohabiting in the device. In this paper, graph theory appears and particularly the study of directed hypergraphs, which helps in the research concerning minimum cost multistandard designs. A cost function which calculates the cost of any possible option of implementation is mentioned but its derivations won't be in the scope of this paper. Our objective is to optimize this proposed cost function to its minimum possible value and thus solving the optimization problem that finds balance between flexibility and computing efficiency. For this, we propose a Minimum Cost Design (MCD) algorithm capable of selecting the option which has the minimum cost to pay and will present its complete set of instructions in this paper. This algorithm exploits various definitions and notations of directed hypergraphs

A Linear Kernel for Planar Vector Domination

Given a graph $G$, an integer $k\geq 0$, and a non-negative integral function $f:V(G) \rightarrow \mathcal{N}$, the {\sc Vector Domination} problem asks whether a set $S$ of vertices, of cardinality $k$ or less, exists in $G$ so that every vertex $v \in V(G)-S$ has at least $f(v)$ neighbors in $S$. The problem generalizes several domination problems and it has also been shown to generalize Bounded-Degree Vertex Deletion. In this paper, the parameterized version of Vector Domination is studied when the input graph is planar. A linear problem kernel is presented

Mathematical Creativity: The Unexpected Links

Creativity in mathematics is identified in many forms or we can say is made up of many components. One of these components is The Unexpected Links where one tries to solve a mathematical problem in a nontraditional manner that requires the formation of hidden bridges between distinct mathematical domains or even between seemingly far ideas within the same domain. In this article, we design problems that express unexpected links in mathematics and suit students of intermediate and secondary levels. We prove their feasibility through teachers’ testimonies and through introducing them in classrooms and collecting students’ attitudes with respect to understanding and interest. Results confirm that students can sense such component and that designed problems had caught teachers’ and students’ interest