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On the degree splitting graph of a graph
Abstract
In this communication we define degree splitting graph of a graph and we study some properties of degree splitting graph.
... The minimum weight of a kRDF is called the k-rainbow domination number of G and is denoted by γ rk (G). In 2004, R. Ponraj and S Somasundaram defined degree splitting graph [2]. ...
... E. J. Cockayne et al. [5] showed that γ(G) ≤ γ R (G) ≤ 2γ(G), for any connected graph G with p vertices. Clearly by [2] we obtain, ...
Consider a graph G(V, E) with vertex partition V = S 1 ∪ S 2 ∪. .. , ∪S t ∪ T where each S i is a set with minimum two vertices having the same degree and T = V \ ∪S i. The degree splitting graph DS(G) is obtained from G by adding vertices w 1 , w 2 ,. .. , w t and joining w i to each vertex of S i (1 ≤ i ≤ t). In this research article we characterize roman domination number of degree splitting graph γ R (DS(G)) and we obtain roman domination number and k-rainbow domination number of degree splitting graphs. Also we establish many bounds on γ R (DS(G)) and γ rk (DS(G)) in terms of elements of G.
... Sampathkumar E and Walikar H B introduced the notion of the splitting graph of a graph in [12]. Ponraj R and S Somasundaram developed the concept of degree splitting of graphs in [9]. C David Raj, K Sunitha and A Subramanian found the radio odd mean and even mean labeling of some graphs in [5]. ...
Radio Even Mean Graceful Labeling of a connected graph G is a bijection from the vertex set V(G) to satisfying the condition for every . A graph which admits radio even mean graceful labeling is called radio even mean graceful graph. In this paper we investigate the radio even mean graceful labeling on degree splitting of some special graphs.
... Recall here that, if G is a graph of set of vertices V(G) = {v 1 , . . . , v n }, and V k (G) ⊆ V(G) denotes its subset of vertices of degree k for all k > 1, then the degree splitting graph [36] (from here on, DSG) of G is the graph DS (G) that results after connecting a new vertex to all those vertices in V k (G), whenever |V k (G)| > 1. From here on, we denote this new vertex by w k , and we say that it constitutes a control node of the initial network. ...
Dynamic coloring has recently emerged as a valuable tool to optimize cryptographic protocols based on secret sharing, which enforce data security in communication networks, and have significant importance in both online storage and cloud computing. This type of graph labeling enables the dealer to distribute secret shares among the nodes of a communication network so that everybody can recover the secret after a minimum number of rounds of communication. This paper delves into this topic by dealing with the dynamic coloring problem for degree splitting graphs. The topological structure of the latter enables the dealer to avoid dishonesty by adding control nodes that supervise all those participants with a similar influence in the network. More precisely, we solve the dynamic coloring problem for degree splitting graphs of any regular graph. The irregular case is partially solved by establishing a lower bound for the corresponding dynamic chromatic number. As illustrative examples, we solve the dynamic coloring problem for the degree splitting graphs of cycles, cocktail, book, comb, fan, jellyfish, windmill and barbell graphs.
... The detour global domination number, denoted byγ d (G) is the minimum cardinality of a detour global dominating set of G and the detour global dominating set with cardinalityγ d (G) is called theγ d -set of G orγ d (G)-set. [6] In [7], R. Ponraj and S. Somasundaram have initiated a study on degree spliting graph DS(G) of a graph G which is defined as follows: ...
In this paper, we introduced the new concept detour global domination number for degree splitting graph of standard graphs. A set S is called a detour global dominating set of G = (V, E) if S is both detour and global dominating set of G. The detour global domination number is the minimum cardinality of a detour global dominating set in G. Let V (G) be S 1 ∪ S 2 ∪ · · · ∪ S t ∪ T, where S i is the set having at least two vertices of same degree and T = V (G) − ∪S i , where 1 ≤ i ≤ t. The degree splitting graph DS(G) is obtained from G by adding vertices w 1 , w 2 , · · · , w t and joining w i to each vertex of S i for i = 1, 2, · · · , t. In this article we recollect the concept of degree splitting graph of a graph and we produced some results based on the detour global domination number for degree splitting graph of path graph, cycle graph, star graph, bistar graph, complete bipartite graph and complete graph.
... A graph which admits 3-equitable labeling is called a 3-equitable graph". Definition 1.2 [9] "Let G be with V = S 1 ∪ S 2 ∪ · · · ∪ S i ∪ T , where each S i is a set of nodes having minimum 2 nodes of the same degree and T = V \ S i . The degree splitting graph of G denoted by DS(G) is derived from G by adding nodes w 1 , w 2 , . . . ...
Let be a simple graph and be an ordered set and. The representation of with respect to is the -tuple . Then is called a monophonic resolving set if different vertices of have different representations with respect to . A monophonic resolving set of minimum number of elements is called a minimum monophonic set for and its cardinality is known as the monophonic metric dimension of , represented by In this article, we determined the monophonic metric dimension of degree splitting graph.
This paper affords a identify and discover human faces is an photograph irrespective of their
position, scale, in-aircraft rotation, orientation, pose, illumination etc. To efficaciously become
aware of the individual through surprising the history and different noises within side the
photograph. It have to be easy and powerful for the customers and become aware of faces
which might be covered with scarf, mask, shades etc, is one of the vital components that
influences the overall performance of face reputation. Many algorithms and technology are
proposed to clear up the occluded face reputation wherein we use haar cascade set of rules that
is simplest one. Initially this set of rules wishes loads of nice pix(pix of faces) and terrible
pix(pix without faces) to teach the classifier then we should extract features from it. Facial
detection is prompted through readability of the photograph, colored or black and white pix. It
can simplest help frontal detection of pix and the education does takes loads of time with a
purpose to separate a terrible face from a terrible face from a nice face.
Keywords—Haar cascade, Face Detection, Recognition, nice face, Terrible fac
The field of graph theory, specifically graph labeling is used in communication networks, particularly in satellite communication. An allocation of numbers to the nodes of a graph under some conditions is a node labeling of. In a “mobile satellite service (MSS)” system, “channel using efficiency is still the main factor” because the systems frequency reuse factor cannot reach 1 exactly. So, researchers have proposed a channel reallocation method using the concept of matching theory to handle this situation by reducing the interference level. A “divisor 3-equitable labeling is a bijective such that i, j ≤ 2, where is the count of lines labeled with i under d. A graph that accepts divisor 3-equitable labeling is called a divisor 3-equitable graph”. This article shows the “existence and non-existence of divisor 3-equitable labeling of certain graphs, besides recalling a few applications of graph labeling in communication networks”.
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