In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k , the k-tuple domination problem is to find a minimum sized vertex subset such that every vertex in the graph is dominated by at least k vertices in this set. The present paper studies the k-tuple domination problem in graphs from an algorithmic point of view. In particular, we give a line...

For a graph G and a vertex-coloring c : V (G) → {1, 2, . . . , k}, the color code of a vertex v is the (k + 1)-tuple (a0, a1, . . . , ak), where a0 = c(v), and for 1 ≤ i ≤ k, ai is the number of neighbors of v colored i. A recognizable coloring is a coloring such that distinct vertices have distinct color codes. The recognition number of a graph is the minimum k for which G has a recognizable k...

Let k ≥ 1 be an integer, and let G be a finite and simple graph with vertex set V (G). A signed total Italian k-dominating function (STIkDF) on a graph G is a functionf : V (G) → {−1, 1, 2} satisfying the conditions that $sum_{xin N(v)}f(x)ge k$ for each vertex v ∈ V (G), where N(v) is the neighborhood of $v$, and each vertex u with f(u)=-1 is adjacent to a vertex v with f(v)=2 or to two vertic...

Let m ≥ 2 and k ≥ 2 be integers and let R be a commutative ring with a unit element denoted by 1. A k-th power diophantine m-tuple in R is an m-tuple (a1, a2, . . . , am) of non-zero elements of R such that aiaj +1 is a k-th power of an element of R for 1 ≤ i < j ≤ m. In this paper, we investigate the case when k ≥ 3 and R = K[X], the ring of polynomials with coefficients in a field K of charac...

Wireless sensor networks propound an algorithmic research problems for prolonging life of nodes and network. The domination algorithms can address some of fundamental issues related to lifetime problems in ad hoc and sensor networks. Most of the graph domination problems are NP-complete even with unit-disk-graphs. The investigation of the thesis addresses some of lifetime issues in sensor netwo...

The VC-dimension of a family P of n-permutations is the largest integer k such that the set of restrictions of the permutations in P on some k-tuple of positions is the set of all k! permutation patterns. Let rk(n) be the maximum size of a set of n-permutations with VC-dimension k. Raz showed that r2(n) grows exponentially in n. We show that r3(n) = 2 Θ(nα(n)) and for every t ≥ 1, we have r2t+2...

The VC-dimension of a family P of n-permutations is the largest integer k such that the set of restrictions of the permutations in P on some k-tuple of positions is the set of all k! permutation patterns. Let rk(n) be the maximum size of a set of n-permutations with VCdimension k. Raz showed that r2(n) grows exponentially in n. We show that r3(n) = 2 Θ(n logα(n)) and for every t ≥ 1, we have r2...

The following generalisation of the Erdős Unit Distance problem was recently suggested by Palsson, Senger, and Sheffer. For a fixed sequence δ = (δ1, …, δk) k distances, (k + 1)-tuple (p1, pk+1) distinct points in ℝd is called k-chain if ∥pj − pj+1∥ δj for every 1 ≤ j k. What maximum number C (n) k-chains set n ℝd? Improving results Sheffer, we essentially determine this all planar case. It onl...

If, for a subset S of Z, we compare the conditions of being parametrizable (a) by a single k-tuple of polynomials with integer coefficients, (b) by a single k-tuple of integer-valued polynomials and (c) by finitely many k-tuples of polynomials with integer coefficients (variables ranging through the integers in each case), then a ⇒ b (obviously), b ⇒ c, and neither implication is reversible. We...