Objectives: To find the connected restrained detour number for standard graphs and mesh graphs. Methods: By determining set with minimum cardinality, of a graph is investigated. Findings: We study that altered when we add pendent vertices. The maximum degree vertices are deleted computed. Novelty: Finding path plays vital role in network-based systems. Planning largest route essential business,...

The domatic number of a graph G, denoted dom(G), is the maximum possible cardinality of a family of disjoint sets of vertices of G, each set being a dominating set of G. It is well known that every graph without isolated vertices has dom(G) ≥ 2. For every k, it is known that there are graphs with minimum degree at least k and with dom(G) = 2. In this paper we prove that this is not the case if ...

Let D be a finite and simple digraph with vertex set V (D), and let f : V (D)→ {−1,1} be a two-valued function. If k ≥ 1 is an integer and ∑x∈N−[v] f (x) ≥ k for each v∈V (D), where N−[v] consists of v and all vertices of D from which arcs go into v, then f is a signed k-dominating function on D. A set { f1, f2, . . . , fd} of distinct signed k-dominating functions of D with the property that ∑...

We investigate the theoretical feasibility of near-optimal, distributed sleep scheduling in energyconstrained sensor networks with pairwise sensor redundancy. In this setting, an optimal sleep schedule is equivalent to an optimal fractional domatic partition of the associated redundancy graph. We present a set of realistic assumptions on the structure of the communication and redundancy relatio...

Let D be a simple digraph with vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and ∑x∈N−[v] f(x) ≥ k for each v ∈ V (D), where N[v] consists of v and all vertices of D from which arcs go into v, then f is a signed k-dominating function on D. A set {f1, f2, . . . , fd} of distinct signed k-dominating functions on D with the property that ∑d i=1 fi(x...

A {em Roman dominating function} on a graph $G$ is a function$f:V(G)rightarrow {0,1,2}$ satisfying the condition that everyvertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex$v$ for which $f(v) =2$. {color{blue}A {em restrained Roman dominating}function} $f$ is a {color{blue} Roman dominating function if the vertices with label 0 inducea subgraph with no isolated vertex.} The wei...

A pattern recognition system based on the n-tuple technique is developed and evaluated for use in classifying non-deterministic data with particular reference to non-restrained hand-written India numerals.
 The presented this research fulfills requirements of simplicity efficiency making it attractive practical present day industrial environments. This operation afforded by self evolving n...

Let k ≥ 1 be an integer. A signed Roman k-dominating function on a digraph D is a function f : V (D) −→ {−1, 1, 2} such that ∑x∈N−[v] f(x) ≥ k for every v ∈ V (D), where N−[v] consists of v and all in-neighbors of v, and every vertex u ∈ V (D) for which f(u) = −1 has an in-neighbor w for which f(w) = 2. A set {f1, f2, . . . , fd} of distinct signed Roman k-dominating functions on D with the pro...

Let D be a finite and simple digraph with vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and ∑ x∈N−[v] f(x) ≥ k for each v ∈ V (D), where N−[v] consists of v and all vertices of D from which arcs go into v, then f is a signed k-dominating function on D. A set {f1, f2, . . . , fd} of distinct signed k-dominating functions of D with the property tha...