Petersen Graph Research Articles

On 2-Rainbow Domination Number of Generalized Petersen Graphs P(5k,k)

We obtain new results on 2-rainbow domination number of generalized Petersen graphs P(5k,k). In some cases (for some infinite families), exact values are established, and in all other cases lower and upper bounds are given. In particular, it is shown that, for k>3, γr2(P(5k,k))=4k for k≡2,8mod10, γr2(P(5k,k))=4k+1 for k≡5,9mod10, 4k+1≤γr2(P(5k,k))≤4k+2 for k≡1,6,7mod10, and 4k+1≤γr2(P(5k,k))≤4k+3 for k≡0,3,4mod10.

The r-Extra Diagnosability of Hyper Petersen Graphs

The diagnosability of a multiprocessor system or an interconnection network plays an important role in measuring the fault tolerance of the network. In 2016, Zhang et al. proposed a new measure for fault diagnosis of the system, namely, the [Formula: see text]-extra diagnosability, which restrains that every fault-free component has at least [Formula: see text] fault-free nodes. As a famous topology structure of interconnection networks, the hyper Petersen graph [Formula: see text] has many good properties. It is difficult to prove the [Formula: see text]-extra diagnosability of an interconnection network. In this paper, we show that the [Formula: see text]-extra diagnosability of [Formula: see text] is [Formula: see text] for [Formula: see text] and [Formula: see text] in the PMC model and for [Formula: see text] and [Formula: see text] in the MM[Formula: see text] model.

Double Roman Domination in Generalized Petersen Graphs

The double Roman domination can be described as a strengthened defense strategy. In an empire, each city can be protected by at most three troops. Every city having no troops must be adjacent to at least two cities with two troops or one city with three troops. Every city having one troop must be adjacent to at least one city with more than one troop. Such an assignment is called a double Roman dominating function (DRDF) of an empire/a graph. The minimum number of troops under such an assignment is the double Roman domination number, denoted as $$\gamma _{dR}$$ . Shao et al. (2018) determine the exact value of $$\gamma _{dR}(P(n,1))$$ . Jiang et al. (2018) determine $$\gamma _{dR}(P(n,2))$$ . In this article, we investigate the double Roman domination number of P(n, k) for $$k\ge 3$$ . We determine the exact value of $$\gamma _{dR}(P(n,k))$$ for $$n\equiv 0(\bmod 4)$$ and $$k\equiv 1(\bmod 2)$$ , and present an improved upper bound of $$\gamma _{dR}(P(n,k))$$ for $$n\not \equiv 0(\bmod 4)$$ or $$k\not \equiv 1(\bmod 2)$$ . Our results imply P(n, 3) for $$n\equiv 0(\bmod 4)$$ is double Roman which can partially answer the open question present by Beeler et al. (2016).

The domination game played on diameter 2 graphs

Let $$\gamma _g(G)$$ be the game domination number of a graph G. It is proved that if $$\mathrm{diam}(G) = 2$$ , then $$\gamma _g(G) \le \left\lceil \frac{n(G)}{2} \right\rceil - \left\lfloor \frac{n(G)}{11}\right\rfloor $$ . The bound is attained: if $$\mathrm{diam}(G) = 2$$ and $$n(G) \le 10$$ , then $$\gamma _g(G) = \left\lceil \frac{n(G)}{2} \right\rceil $$ if and only if G is one of seven sporadic graphs with $$n(G)\le 6$$ or the Petersen graph, and there are exactly ten graphs of diameter 2 and order 11 that attain the bound.

On δ(k)-coloring of generalized Petersen graphs

The chromatic number, [Formula: see text] of a graph [Formula: see text] is the minimum number of colors used in a proper coloring of [Formula: see text]. In an improper coloring, an edge [Formula: see text] is bad if the colors assigned to the end vertices of the edge is the same. Now, if the available colors are less than that of the chromatic number of graph [Formula: see text], then coloring the graph with the available colors leads to bad edges in [Formula: see text]. In this paper, we use the concept of [Formula: see text]-coloring and determine the number of bad edges in generalized Petersen graph ([Formula: see text]). The number of bad edges which result from a [Formula: see text]-coloring of [Formula: see text] is denoted by [Formula: see text].

On cyclic decompositions of the complete graph into the bipartite generalized Petersen graph [formula omitted

A uniformly-ordered ρ-labeling (also known as a ρ++-labeling) of a bipartite graph was introduced by El-Zanati, Vanden Eynden, and Punnim. Such a labeling of a bipartite graph G with m edges yields a cyclic G-decomposition of K2mt+1 for every positive integer t. Here we show that for every even integer n≥8, the generalized Petersen graph P(n,3) admits a ρ++-labeling and hence cyclically decomposes K6nt+1 for all positive integers t.

A new achievement from reviewing some of the content about the edge-distance-balanced property of the generalized Petersen graphs GP(4i + 7, 2)

In this paper, we initially introduce the concept of edge-distance-balanced property which is considered as the generalized concept of distance-balanced property. In our consideration, we will discover some interesting properties of generalized Petersen graphs that called edge distance-balanced graphs. We also define a connection between edge-distance-balanced graphs and distance-balanced graphs. Furthermore, a graph G is said to be edge-distance-balanced if for any edge uv of G, the number of edges closer to u than to v is equal to the number of edges closer to v than to u. It will be concluded that for any round number i ≥ 3, the generalized Petersen graph GP(4i + 7, 2) is not edge-distance-balanced.

Application of Petersen graph pattern technique for automated detection of heart valve diseases with PCG signals

This work aimed to use machine learning to diagnose four heart valve disease conditions and normal heart sounds. This paper proposed the automated classification of normal, aortic stenosis, mitral valve prolapse, mitral stenosis, and mitral regurgitation using phonocardiogram (PCG) signals.This work proposed a novel graph-based feature generator developed using a graph-based technique called Petersen graph pattern (PGP). In addition, a new decomposition model was proposed using variable-sized overlapping blocks, namely tent pooling (TEP) decomposition. By combining TEP and PGP, a novel multilevel feature generation network was developed. Iterative neighborhood component analysis (INCA) was used to select the features. The selected features were fed to decision tree (DT), linear discriminant (LD), bagged tree (BT), and support vector machine (SVM) classifiers for automated classification into five classes.The proposed method's results yielded 100.0% classification accuracy using the k nearest neighbor (kNN) classifier with a ten-fold cross-validation strategy in classifying the five classes. DT, LD, BT, SVM classifiers yielded an accuracy of 95.10%, 98.30%, 98.60%, and 99.90%, respectively.Our attained high classification accuracy suggests that the proposed PGP and TEP based model can be used for heart sound classification using PCG signals.

Double Roman Graphs in P(3k, k)

A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3} with the properties that if f(u)=0, then vertex u is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and if f(u)=1, then vertex u is adjacent to at least one vertex assigned 2 or 3. The weight of f equals w(f)=∑v∈Vf(v). The double Roman domination number γdR(G) of a graph G is the minimum weight of a double Roman dominating function of G. A graph is said to be double Roman if γdR(G)=3γ(G), where γ(G) is the domination number of G. We obtain the sharp lower bound of the double Roman domination number of generalized Petersen graphs P(3k,k), and we construct solutions providing the upper bounds, which gives exact values of the double Roman domination number for all generalized Petersen graphs P(3k,k). This implies that P(3k,k) is a double Roman graph if and only if either k≡0 (mod 3) or k∈{1,4}.

Eternal Domination of Generalized Petersen Graph

An eternal dominating set of a graph G is a set of guards distributed on the vertices of a dominating set so that each vertex can be occupied by one guard only. These guards can defend any infinite series of attacks; an attack is defended by moving one guard along an edge from its position to the attacked vertex. We consider the “all guards move” of the eternal dominating set problem, in which one guard has to move to the attacked vertex and all the remaining guards are allowed to move to an adjacent vertex or stay in their current positions after each attack in order to form a dominating set on the graph and at each step can be moved after each attack. The “all guards move model” is called the m -eternal domination model. The size of the smallest m -eternal dominating set is called the m -eternal domination number and is denoted by γ m ∞ G . In this paper, we find γ m ∞ P n , 1 and γ m ∞ P n , 3 for n ≡ 0 mod 4 . We also find upper bounds for γ m ∞ P n , 2 and γ m ∞ P n , 3 when n is arbitrary.

On the watching number of graphs using discharging procedure

The identifying code has been used to place objects in the sensor and wireless networks. For the vertex x of a graph G, suppose $$N_G[x]$$ is a subset of V(G) containing x and all of its neighbors. Then $$C\subseteq V(G)$$ is called an identifying code of G, if for two distinct vertices x and y of G, both $$C\cap N_G[x]$$ and $$C\cap N_G[y]$$ are nonempty and distinct. The watching system of a graph G has been proposed as an extension of identifying code. In this paper, we study some properties of watching system and then, by using a discharging method, we compute the watching of generalized Petersen graph P(n, k) (for $$k=2,3,n/2$$ ) which is commonly used in interconnection networks.

Ricci-flat cubic graphs with girth five

We classify all connected, simple, 3-regular graphs with girth at least 5 that are Ricci-flat. We use the definition of Ricci curvature on graphs given in Lin-Lu-Yau, Tohoku Math., 2011, which is a variation of Ollivier, J. Funct. Anal., 2009. A graph is Ricci-flat, if it has vanishing Ricci curvature on all edges. We show, that the only Ricci-flat cubic graphs with girth at least 5 are the Petersen graph, the Triplex and the dodecahedral graph. This will correct the classification in Lin-Lu-Yau, Comm. Anal. Geom., 2014, that misses the Triplex.

Decomposition of generalized Petersen graphs into claws, cycles and paths

Let G = (V,E) be a finite graph with n vertices. Let n and k be positive integers where n ≥ 3 and 1 ≤ k < n/2. The Generalized Petersen Graph GP(n,k) is a graph with vertex set {u 0 ,u 1 ,u 2 ,...,u n−1 , v 0 , v 1 , v 2 ,..., v n−1 } and edge-set consisting of all edges of the form u i u i+1 ,uivi and v i v i+k where 0 ≤ i ≤ n−1, the subscripts being reduced modulo n. Obviously GP(n, k) is always a cubic graph and GP(5,2) is the well-known Petersen graph. In this paper, we show that GP(n,1), n ≥ 3 can be decomposed into n copies of S 3 if n is even and P 4 and (n−1) copies of S 3 if n is odd. Also, we show that GP(n,2), n ≥ 5 can be decomposed into n/2 copies of S3, 2 copies of C n/2 and n/2 copies of P 2 if n is even and C n , P 4 , [n/2] copies of S 3 and (n/2 −1) copies of P 2 if n is odd. GP(n,2), n ≥ 5 and n = 3d, d = 2,3,... can be decomposed into 2d copies of S3 and d copies of P4. GP(n,2), n ≥ 5 and n = 4d, d = 2,3,... can be decomposed into 3d copies of S 3 and d copies of P 4 . GP(n,3), n ≥ 8 can be decomposed into n copies of S 3 if n is even and P 6 , P 2 and (n−2) copies of S 3 if n is odd.

The Petersen and Heawood graphs make up graphical twins via induced matchings

The Petersen and Heawood graphs make up graphical twins via induced matchings

Injective edge coloring of generalized Petersen graphs

<abstract> Three edges $ e_1 $, $ e_2 $ and $ e_3 $ in a graph $ G $ are $ consecutive $ if they form a cycle of length $ 3 $ or a path in this order. A $ k $-$ injective\; edge\; coloring $ of a graph $ G $ is an edge coloring of $ G $, (not necessarily proper), such that if edges $ e_1 $, $ e_2 $, $ e_3 $ are consecutive, then $ e_1 $ and $ e_3 $ receive distinct colors. The minimum $ k $ for which $ G $ has a $ k $-injective edge coloring is called the $ injective\; edge\; coloring\; number $, denoted by $ \chi_i^{\prime}(G) $. In this paper, we consider the injective edge coloring numbers of generalized Petersen graphs $ P(n, 1) $ and $ P(n, 2) $. We determine the exact values of injective edge coloring numbers for $ P(n, 1) $ with $ n\geq 3 $, and for $ P(n, 2) $ with $ 4\leq n\leq 7 $. For $ n\geq 8 $, we show that $ 4\leq \chi_{i}^{'}(P(n, 2))\leq 5. $ </abstract>

On the neighbor-distinguishing in generalized Petersen graphs

&lt;abstract&gt;&lt;p&gt;In a connected graph $ G $, two adjacent vertices are said to be neighbors of each other. A vertex $ v $ adjacently distinguishes a pair $ (x, y) $ of two neighbors in $ G $ if the number of edges in $ v $-$ x $ geodesic and the number of edges in $ v $-$ y $ geodesic differ by one. A set $ S $ of vertices of $ G $ is a neighbor-distinguishing set for $ G $ if every two neighbors in $ G $ are adjacently distinguished by some element of $ S $. In this paper, we consider two families of generalized Petersen graphs and distinguish every two neighbors in these graphs by investigating their minimum neighbor-distinguishing sets, which are of coordinately two.&lt;/p&gt;&lt;/abstract&gt;

Petersen Graph Multi-Orientation Based Multi-Scale Ternary Pattern (PGMO-MSTP): An Efficient Descriptor for Texture and Material Recognition

Classifying and modeling texture images, especially those with significant rotation, illumination, scale, and view-point variations, is a hot topic in the computer vision field. Inspired by local graph structure (LGS), local ternary patterns (LTP), and their variants, this paper proposes a novel image feature descriptor for texture and material classification, which we call Petersen Graph Multi-Orientation based Multi-Scale Ternary Pattern (PGMO-MSTP). PGMO-MSTP is a histogram representation that efficiently encodes the joint information within an image across feature and scale spaces, exploiting the concepts of both LTP-like and LGS-like descriptors, in order to overcome the shortcomings of these approaches. We first designed two single-scale horizontal and vertical Petersen Graph-based Ternary Pattern descriptors ( PGTPh and PGTPv ). The essence of PGTPh and PGTPv is to encode each 5×5 image patch, extending the ideas of the LTP and LGS concepts, according to relationships between pixels sampled in a variety of spatial arrangements (i.e., up, down, left, and right) of Petersen graph-shaped oriented sampling structures. The histograms obtained from the single-scale descriptors PGTPh and PGTPv are then combined, in order to build the effective multi-scale PGMO-MSTP model. Extensive experiments are conducted on sixteen challenging texture data sets, demonstrating that PGMO-MSTP can outperform state-of-the-art handcrafted texture descriptors and deep learning-based feature extraction approaches. Moreover, a statistical comparison based on the Wilcoxon signed rank test demonstrates that PGMO-MSTP performed the best over all tested data sets.

Pronic Heron Mean labeling on special cases of generalized Peterson graph $ P(n,k) $ and disconnected graphs

Pronic Heron Mean labeling on special cases of generalized Peterson graph $ P(n,k) $ and disconnected graphs

The K -Size Edge Metric Dimension of Graphs

In this paper, a new concept k -size edge resolving set for a connected graph G in the context of resolvability of graphs is defined. Some properties and realizable results on k -size edge resolvability of graphs are studied. The existence of this new parameter in different graphs is investigated, and the k -size edge metric dimension of path, cycle, and complete bipartite graph is computed. It is shown that these families have unbounded k -size edge metric dimension. Furthermore, the k-size edge metric dimension of the graphs Pm □ Pn, Pm □ Cn for m, n ≥ 3 and the generalized Petersen graph is determined. It is shown that these families of graphs have constant k -size edge metric dimension.

Weakly convex and convex domination numbers for generalized Petersen and flower snark graphs

We consider the weakly convex and convex domination numbers for two classes of graphs: generalized Petersen graphs and flower snark graphs. For a given generalized Petersen graph GP(n, k), we prove that if k = 1 and n ≥ 4 then both the weakly convex domination number γwcon(GP(n, k)) and the convex domination number γcon(GP(n, k)) are equal to n. For k ≥ 2 and n ≥ 13, γwcon(GP(n, k)) = γcon(GP(n, k)) = 2n, which is the order of GP(n, k). Special cases for smaller graphs are solved by the exact method. For a flower snark graph Jn, where n is odd and n ≥ 5, we prove that γwcon(Jn) =2n and γcon(Jn) = 4n.