Showing papers on "Path graph published in 2022"

The existence of path-factor uniform graphs with large connectivity

Abstract: A path-factor is a spanning subgraph F of G such that every component of F is a path with at least two vertices. Let k ≥ 2 be an integer. A P ≥ k -factor of G means a path factor in which each component is a path with at least k vertices. A graph G is a P ≥ k -factor covered graph if for any e ∈ E ( G ), G has a P ≥ k -factor covering e . A graph G is called a P ≥ k -factor uniform graph if for any e 1 ,e 2 ∈ E ( G ) with e 1 6= e 2 , G has a P ≥ k -factor covering e 1 and avoiding e 2 . In other words, a graph G is called a P ≥ k -factor uniform graph if for any e ∈ E ( G ), G − e is a P ≥ k -factor covered graph. In this paper, we present two sufficient conditions for graphs to be P ≥3 -factor uniform graphs depending on binding number and degree conditions. Furthermore, we show that two results are best possible in some sense.

GGraph: An Efficient Structure-Aware Approach for Iterative Graph Processing

Abstract: Many iterative graph processing systems have recently been developed to analyze graphs. Although they are effective from different aspects, there is an important issue that has not been addressed yet. A real-world graph follows the power-law property, in which a small number of vertices have high degrees (i.e., are connected to most other vertices in the graph). These vertices are called hot-vertices and usually require more iterations to converge. In the existing solutions, these hot-vertices may be allocated to many or even all graph partitions along with other vertices that are easy to converge. As the result, the partitions with hot-vertices have to be loaded repeatedly (and consequently the system suffers from high data access cost), although perhaps only a few vertices in these partitions are active. To cope with this issue, we develop an efficient open source graph partition manager, called GGraph, which can be integrated into the existing graph processing systems to efficiently support iterative graph processing, by taking into account the power-law property of the graph structure. It uses a novel graph repartitioning scheme with low overhead to dynamically partition the hot-vertices together, so as to avoid loading the inactive vertices in the same partition as the repeatedly processed hot-vertices. By such means, it not only enables less data access cost, but also enables the privileged processing of the hot-vertices. In order to further increase the convergence speed, a scheduling algorithm is further proposed in this work to prioritize the processing of the hot-vertices with low overhead. To demonstrate the efficiency of GGraph, we plug it into four state-of-the-art graph processing systems, i.e., Gemini, GraphChi, Chaos, and GridGraph, and experimental results show that GGraph improves their performance by up to 3.2 times, 3.8 times, 3.9 times, 3.5 times, respectively.

Review of Shortest Path Algorithm

Abstract: Shortest path algorithm. Graphs are an example of non-linear data structure. A graph is a collection of nodes which are connected by edges. The definition of graph G = (V, E) is basically a collection of vertices and edges. Graphs can be classified on the basis of types of edges. Directed graphs have each of the edges directed which means the edges connecting the two nodes defines the way it is connected from and to. On the other side, undirected graphs have edges which have no direction. The edges of a graph have weights which are associated with it. The weight of an edge can be thought as the cost of the edge. Let’s assume there are two vertices representing two cities, then the weight of the edge between the vertices may represent the distance between the cities. Given a given graph and a particular node, we can find a path of least total weight from that node to other vertices of the graph. The total weight of the path will be the sum of the weights of the edges. Graphs can be used in real life to find the shortest path between two destinations, used in social networking sites like facebook and the world wide web where the web pages are represented by the nodes. Dijkstra’s Algorithm, Floyd – Warshall, Bellman Ford Algorithm, Johnson’s algorithm, A* search algorithm are some of the shortest path algorithms.

On joint properties of vertices with a given degree or label in the random recursive tree

Abstract: . In this paper, we study the joint behaviour of the degree, depth and label of and graph distance between high-degree vertices in the random recursive tree. We generalise the results obtained by Eslava [12] and extend these to include the labels of and graph distance between high-degree vertices. The analysis of both these two properties of high-degree vertices is novel, in particular in relation to the behaviour of the depth of such vertices. In passing, we also obtain results for the joint behaviour of the degree and depth of and graph distance between any fixed number of vertices with a prescribed label. This combines several isolated results on the degree [22], depth [7, 24] and graph distance [9, 15] of vertices with a prescribed label already present in the literature. Furthermore, we extend these results to hold jointly for any number of fixed vertices and improve these results by providing more detailed descriptions of the distributional limits. Our analysis is based on a correspondence between the random recursive tree known and a representation of the Kingman n -coalescent.

A Structured Optimal Controller With Feed-Forward for Transportation

Abstract: We study an optimal control problem for a simple transportation model on a path graph. We give a closed form solution for the optimal controller, which can also account for planned disturbances using feed-forward. The optimal controller is highly structured, which allows the controller to be implemented using only local communication, conducted through two sweeps through the graph.

k-Path-Connectivity of Completely Balanced Tripartite Graphs

Abstract: For a graph G=(V,E) and a set S⊆V(G) of a size at least 2, a path in G is said to be an S-path if it connects all vertices of S. Two S-paths P1 and P2 are said to be internally disjoint if E(P1)∩E(P2)=∅ and V(P1)∩V(P2)=S; that is, they share no vertices and edges apart from S. Let πG(S) denote the maximum number of internally disjoint S-paths in G. The k-path-connectivity πk(G) of G is then defined as the minimum πG(S), where S ranges over all k-subsets of V(G). In this paper, we study the k-path-connectivity of the complete balanced tripartite graph Kn,n,n and obtain πkKn,n,n=2nk−1 for 3≤k≤n.

Perfect State Transfer in Arbitrary Distance

Abstract: Quantum Perfect State Transfer (PST) is a fundamental tool of quantum communication in a network. It is considered a rare phenomenon. The original idea of PST depends on the fundamentals of the continuous-time quantum walk. A path graph with at most three vertices allows PST. Based on the Markovian quantum walk, we introduce a significantly powerful method for PST in this article. We establish PST between the extreme vertices of a path graph of arbitrary length. Moreover, any pair of vertices $j$ and $n - j - 1$ in a path graph with $n$ vertices allow PST for $0 \leq j < \frac{n - 1}{2}$. Also, no cycle graph with more than $4$ vertices does not allow PST based on the continuous-time quantum walk. In contrast, we establish PSTs based on Markovian quantum walk between the pair of vertices $j$ and $j + m$ for $j = 0, 1, \dots (m - 1)$ in a cycle graph with $2m$ vertices.

Computing and Listing Avoidable Vertices and Paths

Abstract: A simplicial vertex of a graph is a vertex whose neighborhood is a clique. It is known that listing all simplicial vertices can be done in O(nm) time or $$O(n^{\omega })$$ time, where $$O(n^{\omega })$$ is the time needed to perform a fast matrix multiplication. The notion of avoidable vertices generalizes the concept of simplicial vertices in the following way: a vertex u is avoidable if every induced path on three vertices with middle vertex u is contained in an induced cycle. We present algorithms for listing all avoidable vertices of a graph through the notion of minimal triangulations and common neighborhood detection. In particular we give algorithms with running times $$O(n^{2}m)$$ and $$O(n^{1+\omega })$$ , respectively. Additionally, based on a simplified graph traversal we propose a fast algorithm that runs in time $$O(n^2 + m^2)$$ and matches the corresponding running time of listing all simplicial vertices on sparse graphs with $$m=O(n)$$ . Moreover, we show that our algorithms cannot be improved significantly, as we prove that under plausible complexity assumptions there is no truly subquadratic algorithm for recognizing an avoidable vertex. To complement our results, we consider their natural generalizations of avoidable edges and avoidable paths. We propose an O(nm)-time algorithm that recognizes whether a given induced path is avoidable.

A combinatorial bound on the number of distinct eigenvalues of a graph

Abstract: The smallest possible number of distinct eigenvalues of a graph $G$, denoted by $q(G)$, has a combinatorial bound in terms of unique shortest paths in the graph. In particular, $q(G)$ is bounded below by $k$, where $k$ is the number of vertices of a unique shortest path joining any pair of vertices in $G$. Thus, if $n$ is the number of vertices of $G$, then $n-q(G)$ is bounded above by the size of the complement (with respect to the vertex set of $G$) of the vertex set of the longest unique shortest path joining any pair of vertices of $G$. The purpose of this paper is to commence the study of the minor-monotone floor of $n-k$, which is the minimum of $n-k$ among all graphs of which $G$ is a minor. Accordingly, we prove some results about this minor-monotone floor.

Bilangan Terhubung Titik Pelangi Kuat Graf Octa-Chain (OCm)

Abstract: An Octa-Chain graph (OCm) is a graph formed by modifying the cycle graph C8 by adding an edge connecting the midpoints in C8. The minimum number of colors used to color the vertices in a graph so that every two vertices have a rainbow path is called the rainbow vertex-connected number denoted by rvc (G). While the minimum number of colors used to color the vertices in a graph so that every two vertices are always connected by a rainbow path is called a strong rainbow vertex connected number and is denoted by srvc (G). This study aims to determine the rainbow vertex-connected number (rvc) and the strong rainbow-vertex-connected number (srvc) in the Octa-Chain graph (OCm). The results obtained from this research are the rainbow vertex-connected number rvc (OCm)=2m and the strong rainbow-vertex-connected number srvc (OCm)=2m.

On joint properties of vertices with a given degree or label in the random recursive tree

Abstract: In this paper, we study the joint behaviour of the degree, depth and label of and graph distance between high-degree vertices in the random recursive tree. We generalise the results obtained by Eslava and extend these to include the labels of and graph distance between high-degree vertices. The analysis of both these two properties of high-degree vertices is novel, in particular in relation to the behaviour of the depth of such vertices. In passing, we also obtain results for the joint behaviour of the degree and depth of and graph distance between any fixed number of vertices with a prescribed label. This combines several isolated results on the degree and depth of and graph distance between vertices with a prescribed label already present in the literature. Furthermore, we extend these results to hold jointly for any number of fixed vertices and improve these results by providing more detailed descriptions of the distributional limits. Our analysis is based on a correspondence between the random recursive tree and a representation of the Kingman $n$-coalescent.

Kajian Rainbow 2-Connected Pada Graf Eksponensial dan Beberapa Operasi Graf

Abstract: Let $G=(V(G),E(G))$ is a graph connected non-trivial. \textit{Rainbow connection} is edge coloring on the graph defined as $f:E(G)\rightarrow \{1,2,...,r|r \in N\}$, for every two distinct vertices in $G$ have at least one \textit{rainbow path}. The graph $G$ says \textit{rainbow connected} if every two vertices are different in $G$ associated with \textit{rainbow path}. A path $u-v$ in $G$ says \textit{rainbow path} if there are no two edges in the trajectory of the same color. The edge coloring sisi cause $G$ to be \textit{rainbow connected} called \textit{rainbow coloring}. Minimum coloring in a graph $G$ called \textit{rainbow connection number} which is denoted by $rc(G)$. If the graph $G$ has at least two \textit{disjoint rainbow path} connecting two distinct vertices in $G$. So graph $G$ is called \textit{rainbow 2-connected} which is denoted by $rc_2(G)$. The purpose of this research is to determine \textit{rainbow 2-connected} of some resulting graph operations. This research study \textit{rainbow 2-connected} on the graph (${C_4}^{K_n}$ and $Wd_{(3,2)}\square K_n$).

Multimarked Spatial Search by Continuous-Time Quantum Walk

Abstract: The quantum-walk-based spatial search problem aims to find a marked vertex using a quantum walk on a graph with marked vertices. We describe a framework for determining the computational complexity of spatial search by continuous-time quantum walk on arbitrary graphs by providing a recipe for finding the optimal running time and the success probability of the algorithm. The quantum walk is driven by a Hamiltonian that is obtained from the adjacency matrix of the graph modified by the presence of the marked vertices. The success of our framework depends on the knowledge of the eigenvalues and eigenvectors of the adjacency matrix. The spectrum of the modified Hamiltonian is then obtained from the roots of the determinant of a real symmetric matrix, whose dimension depends on the number of marked vertices. We show all steps of the framework by solving the spatial searching problem on the Johnson graphs with fixed diameter and with two marked vertices. Our calculations show that the optimal running time is $O(\sqrt{N})$ with asymptotic probability $1+o(1)$, where $N$ is the number of vertices.

Prime Cordial Labeling of Generalized Petersen Graph under Some Graph Operations

Abstract: A graph is a connection of objects. These objects are often known as vertices or nodes and the connection or relation in these nodes are called arcs or edges. There are certain rules to allocate values to these vertices and edges. This allocation of values to vertices or edges is called graph labeling. Labeling is prime cordial if vertices have allocated values from 1 to the order of graph and edges have allocated values 0 or 1 on a certain pattern. That is, an edge has an allocated value of 0 if the incident vertices have a greatest common divisor (gcd) greater than 1. An edge has an allocated value of 1 if the incident vertices have a greatest common divisor equal to 1. The number of edges labeled with 0 or 1 are equal in numbers or, at most, have a difference of 1. In this paper, our aim is to investigate the prime cordial labeling of rotationally symmetric graphs obtained from a generalized Petersen graph P(n,k) under duplication operation, and we have proved that the resulting symmetric graphs are prime cordial. Moreover, we have also proved that when we glow a Petersen graph with some path graphs, then again, the resulting graph is a prime cordial graph.

Pendant and isolated vertices of comaximal graphs of modules

Abstract: A comaximal graph Γ(M) is an undirected graph with vertex set as the collection of all submodules of a module M and any two vertices A and B are adjacent if and only if A + B = M. We discuss characteristics of pendant vertices in Γ(M). We also observe features of isolated vertices in a special spanning subgraph in Γ(M).

On the unicyclic graphs having vertices that belong to all their (strong) metric bases

Abstract: A metric basis in a graph $G$ is a smallest possible set $S$ of vertices of $G$, with the property that any two vertices of $G$ are uniquely recognized by using a vector of distances to the vertices in $S$. A strong metric basis is a variant of metric basis that represents a smallest possible set $S'$ of vertices of $G$ such that any two vertices $x,y$ of $G$ are uniquely recognized by a vertex $v\in S'$ by using either a shortest $x-v$ path that contains $y$, or a shortest $y-v$ path that contains $x$. Given a graph $G$, there exist sometimes some vertices of $G$ such that they forcedly belong to every metric basis or to every strong metric basis of $G$. Such vertices are called (resp. strong) basis forced vertices in $G$. It is natural to consider finding them, in order to find a (strong) metric basis in a graph. However, deciding about the existence of these vertices in arbitrary graphs is in general an NP-hard problem, which makes desirable the problem of searching for (strong) basis forced vertices in special graph classes. This article centers the attention in the class of unicyclic graphs. It is known that a unicyclic graph can have at most two basis forced vertices. In this sense, several results aimed to classify the unicyclic graphs according to the number of basis forced vertices they have are given in this work. On the other hand, with respect to the strong metric bases, it is proved in this work that unicyclic graphs can have as many strong basis forced vertices as we would require. Moreover, some characterizations of the unicyclic graphs concerning the existence or not of such vertices are given in the exposition as well.

Existence of $2$-Factors in Tough Graphs without Forbidden Subgraphs

Abstract: For a given graph $R$, a graph $G$ is $R$-free if $G$ does not contain $R$ as an induced subgraph. It is known that every $2$-tough graph with at least three vertices has a $2$-factor. In graphs with restricted structures, it was shown that every $2K_2$-free $3/2$-tough graph with at least three vertices has a $2$-factor, and the toughness bound $3/2$ is best possible. In viewing $2K_2$, the disjoint union of two edges, as a linear forest, in this paper, for any linear forest $R$ on 5, 6, or 7 vertices, we find the sharp toughness bound $t$ such that every $t$-tough $R$-free graph on at least three vertices has a 2-factor.

Shortest Beer Path Queries in Outerplanar Graphs

Abstract: A beer graph is an undirected graph G, in which each edge has a positive weight and some vertices have a beer store. A beer path between two vertices u and v in G is any path in G between u and v that visits at least one beer store. We show that any outerplanar beer graph G with n vertices can be preprocessed in O(n) time into a data structure of size O(n), such that for any two query vertices u and v, (i) the weight of the shortest beer path between u and v can be reported in $$O(\alpha (n))$$ time (where $$\alpha (n)$$ is the inverse Ackermann function), and (ii) the shortest beer path between u and v can be reported in O(L) time, where L is the number of vertices on this path. Note that the running time for (ii) does not depend on the number of vertices of G. Both results are optimal, even when G is a beer tree (i.e., a beer graph whose underlying graph is a tree).

Disjoint Edges in Geometric Graphs

Abstract: A geometric graph is a graph drawn in the plane so that its vertices and edges are represented by points in general position and straight line segments, respectively. A vertex of a geometric graph is called pointed if it lies outside of the convex hull of its neighbours. We show that for a geometric graph with $$n$$ vertices and $$e$$ edges there are at least $$\frac{n}{2}\left(\begin{array}{cc}2e/n\\3\end{array}\right)$$ pairs of disjoint edges provided that $$2e\ge n$$ and all the vertices of the graph are pointed. Besides, we prove that if any edge of a geometric graph with $$n$$ vertices is disjoint from at most $$m$$ edges, then the number of edges of this graph does not exceed $$n(\sqrt{1+8m}+3)/4$$ provided that $$n$$ is sufficiently large. These two results are tight for an infinite family of graphs.

On the crossing numbers of the join products of five graphs on six vertices with discrete graph

Abstract: The crossing number $\mathrm{cr}(G)$ of a graph $G$ is the minimum number of edge crossings over all drawings of $G$ in the plane. In the paper, the crossing number of the join product $G^\ast + D_n$ for the connected graph $G^\ast$ on six vertices consisting of one path on four vertices $P_4$ and two leaves adjacent with the same outer vertex of the path $P_4$ is given, where $D_n$ consists of $n$ isolated vertices. Finally, by adding some edges to the graph $G^\ast$, we obtain the crossing numbers of the join products of other four graphs with $D_n$.

Gallai's Path Decomposition for 2-degenerate Graphs

Abstract: Gallai's path decomposition conjecture states that if $G$ is a connected graph on $n$ vertices, then the edges of $G$ can be decomposed into at most $\lceil \frac{n }{2} \rceil$ paths. A graph is said to be an odd semi-clique if it can be obtained from a clique on $2k+1$ vertices by deleting at most $k-1$ edges. Bonamy and Perrett asked if the edges of every connected graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n}{2} \rfloor$ paths unless $G$ is an odd semi-clique. A graph $G$ is said to be 2-degenerate if every subgraph of $G$ has a vertex of degree at most $2$. In this paper, we prove that the edges of any connected 2-degenerate graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n }{2} \rfloor$ paths unless $G$ is a triangle.

Total Magic Cordial and Total Sequential Cordial Labeling of Path Related Graph

Abstract: A graph is simple and undirected and it is defined as G(V,E) with V as vertex set and E as edge set. Path graph is a simple graph whose vertices and edges are denoted as v1,v2,...vn and edges are vivi+1 with n number of vertices and n-1 number of edges. This chapter is discussing the cordial related labeling for special class of path graph called Square graph of path and Shadow graph of path. A graph G is said to have Total Sequential Cordial labeling, if there exists a mapping f : V U E\(\rightarrow \) {0,1} such that for each (a,b) \(\in\) E,f (a,b) =|f(a) - f(b)|, provided the condition |f(0) - f(1)|\(\leq\)1 hold, where f(0) = vf (0) + ef (0) and f(1) = vf (1) + ef (1) and vf (1), ef (1),i \(\in\) {0,1} are respectively, the number of vertices and edges labeled with i. Total magic cordial labeling is defined as ,if there exists a mapping f : V U E\(\rightarrow \) {0,1} such that f(a) + f (b) + f(ab) = Cmod2 for all (a,b) \(\in\) E provided the condition |f(0) - f(1)| \(\leq\) 1 is hold, where f(0) = vf (0) + ef (0) and f(1) = vf (1) + ef (1) and vf (i), ef (i),i \(\in\) {0,1} are respectively, the number of vertices and edges labeled with i.

Enumerate the Number of Vertices Labeled Connected Graph of Order Seven Containing No Parallel Edges

Abstract: A graph that is connected G(V,E) is a graph in which there is at least one path connecting every two vertices in G; otherwise, it is called a disconnected graph. Labels or values can be assigned to the vertices or edges of a graph. A vertex-labeled graph is one in which only the vertices are labeled, and an edges-labeled graph is one in which only edges are assigned values or labels. If both vertices and edges are labeled, the graph is referred to as total labeling. If given n vertices and m edges, numerous graphs can be made, either connected or disconnected. This study will be discussed the number of disconnected vertices labeled graphs of order seven containing no parallel edges and may contain loops. The results show that number of vertices labeled connected graph of order seven with no parallel edges is N(G7,m, g)l= 6,727×Cm6; while for 7≤g≤ 21, N(G7,m, g)l= kg C(m−(g−6))g−1, where k7 =30,160, k8 = 30,765, k9=21,000, k10 =28,364, k11= 26,880, k12=26,460 , k13 = 20,790, k14 =10,290, k15 = 8,022, k16 = 2,940, k17 =4,417, k18 = 2,835, k19 =210, k20 = 21, k21= 1.

The quantum search of many vertices on the joined complete graph

Abstract: The quantum search on the graph is a very important topic. In this work, we develop the theoretic method about the searching single vertex on the graph [Phys. Rev. Lett. 114, 110503 (2015)], and systematically study the search of many vertices on one low-connectivity graph, the joined complete graph. Our results reveal that, with the optimal jumping rate obtained from the theoretic method, we can find such target vertices at the time O√N , where N is the number of total vertices. Therefore, the search of many vertices on the joined complete graph possessing quantum advantage has achieved.

On the maximum spread of planar and outerplanar graphs

Abstract: The spread of a graph $G$ is the difference between the largest and smallest eigenvalue of the adjacency matrix of $G$. Gotshall, O'Brien and Tait conjectured that for sufficiently large $n$, the $n$-vertex outerplanar graph with maximum spread is the graph obtained by joining a vertex to a path on $n-1$ vertices. In this paper, we disprove this conjecture by showing that the extremal graph is the graph obtained by joining a vertex to a path on $\lceil (2n-1)/3\rceil$ vertices and $\lfloor(n-2)/3\rfloor$ isolated vertices. For planar graphs, we show that the extremal $n$-vertex planar graph attaining the maximum spread is the graph obtained by joining two nonadjacent vertices to a path on $\lceil(2n-2)/3\rceil$ vertices and $\lfloor(n-4)/3\rfloor$ isolated vertices.