Let Z2 = {0, 1} and G = (V ,E) be a graph. A labeling f : V → Z2 induces an edge labeling f* : E →Z2 defined by f*(uv) = f(u).f (v). For i ε Z2 let vf (i) = v(i) = card{v ε V : f(v) = i} and ef (i) = e(i) = {e ε E : f*(e) = i}. A labeling f is said to be Vertex-friendly if | v(0) − v(1) |≤ 1. The vertex balance index set is defined by {| ef (0) − ef (1) | : f is vertex-friendly}. In this paper ...

Let P1, P2 be graph properties. A vertex (P1,P2)-partition of a graph G is a partition {V1, V2} of V (G) such that for i = 1, 2 the induced subgraph G[Vi] has the property Pi. A property R = P1◦P2 is defined to be the set of all graphs having a vertex (P1,P2)-partition. A graph G ∈ P1◦P2 is said to be uniquely (P1,P2)-partitionable if G has exactly one vertex (P1,P2)-partition. In this note, we...

The notion of a first order topological structure was introduced by Pillay [1] as a generalization of the notion of an o-minimal structure. The idea is to provide a general framework in which model theory can be used to analyze a topological structure whose topology isn’t necessarily induced by a definable order. In the o-minimal case, the topology is generated from a basis where each basis set...

An operation is called broadcasting if a node sends a packet to all other nodes in an ad hoc network. Broadcasting is an elementary operation to support many applications in ad hoc networks. Thus, many schemes have been proposed for reducing the number of re-broadcasting packets. However, to the best of our knowledge, few papers discuss the bound of necessary broadcasting packets. In the work, ...

We provide polynomial time data reduction rules for Connected Dominating Set in planar graphs and analyze these to obtain a linear kernel for the planar Connected Dominating Set problem. To obtain the desired kernel we introduce a method that we call reduce or refine. Our kernelization algorithm analyzes the input graph and either finds an appropriate reduction rule that can be applied, or zoom...

Let G = (V,E) be a simple undirected graph with n vertices then a set partition π = {V1, . . . , Vk} of the vertex set of G is a connected set partition if each subgraph G[Vj ] induced by the blocks Vj of π is connected for 1 ≤ j ≤ k. Define qi(G) as the number of connected set partitions in G with i blocks. The partition polynomial is Q(G,x) = ∑n i=0 qi(G)x . This paper presents a splitting ap...

Connected Dominating Set is popularly used for constructing virtual backbones for broadcasting operation in WSNs. UD Graph is the most suitable model for a wireless sensor network. In this paper we provide an algorithm to find MCDS in UD Graph. It is based on the computation of convex hulls of sensor nodes or vertices. Constructing a virtual backbone in WSNs is an important issue because it red...

Let L be the space of line transversals to a nite family of pairwise disjoint compact convex sets in R 3. We prove that each connected component of L can itself be represented as the space of transversals to some nite family of pairwise disjoint compact convex sets.

a vertex irregular total k-labeling of a graph g with vertex set v and edge set e is an assignment of positive integer labels {1, 2, ..., k} to both vertices and edges so that the weights calculated at vertices are distinct. the total vertex irregularity strength of g, denoted by tvs(g)is the minimum value of the largest label k over all such irregular assignment. in this paper, we study the to...