Let G = (V, E) be a simple graph. A set S  V is a dominating set of G, if every vertex in V-S is adjacent to at least one vertex in S. Let be the square of the Path and let 2 n P n P   2 , n D P i denote the family of all dominating sets of with cardinality i. Let 2 n P     2 , n n d P i D P i  2 ,  . In this paper, we obtain a recursive formula for . Using this recursive formula, we c...

As a natural variant of domination in graphs, Dankelmann et al. [Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduced exponential domination, where vertices are considered to have some dominating power that decreases exponentially with the distance, and the dominated vertices have to accumulate a sufficient amount of this power emanating from the dominating vertice...

We study a relaxation of the Vector Domination problem called Vector Connectivity (VecCon). Given a graph G with a requirement r(v) for each vertex v, VecCon asks for a minimum cardinality set of vertices S such that every vertex v ∈ V \S is connected to S via r(v) disjoint paths. In the paper introducing the problem, Boros et al. [Networks, 2014, to appear] gave polynomial-time solutions for V...

We consider two general frameworks for multiple domination, which are called 〈r, s〉-domination and parametric domination. They generalise and unify {k}-domination, k-domination, total k-domination and k-tuple domination. In this paper, known upper bounds for the classical domination are generalised for the 〈r, s〉-domination and parametric domination numbers. These generalisations are based on t...

The Asymmetric Traveling Salesman Problem (ATSP) is stated as follows. Given a weighted complete digraph (K∗ n, w), find a Hamilton cycle (called a tour) in K∗ n of minimum cost. Here the weight function w is a mapping from A(K∗ n), the set of arcs in K∗ n, to the set of reals. The weight of an arc xy of K∗ n is w(x, y). The weight w(D) of a subdigraph D of K∗ n is the sum of the weights of arc...

Eternal and m-eternal domination are concerned with using mobile guards to protect a graph against infinite sequences of attacks at vertices. Eternal domination allows one guard to move per attack, whereas more than one guard may move per attack in the m-eternal domination model. Inequality chains consisting of the domination, eternal domination, m-eternal domination, independence, and clique c...

A class domination coloring (also called cd-coloring) of a graph is a proper coloring such that for every color class, there is a vertex that dominates it. The minimum number of colors required for a cd-coloring of the graph G, denoted by χcd(G), is called the class domination chromatic number (cd-chromatic number) of G. In this work, we consider two problems associated with the cd-coloring of ...

BackgroundGame Theory Interpretation MethodsRandomizationFunctional Lagrange Multipliers ConclusionsReferences