Problem In Graph Theory Research Articles

Graph Isomorphism in View of Decomposition Technique

Graph Isomorphism is anopen problem in graph theory. A large number of graphs have the polynomial time algo-rithms for detection. But in general there are specific algorithms for certain range of vertices. Our try to design a generic algo-rithm. In this paper we proposed an algorithm of O(n3) complexi-ty. The decomposition techniques have been adopted as well as reconstruction conjecture in reverse for taking decision of whether the two graphs are isomorphic or not.The ideology is inherited from DNA isolation in molecular biology [Chomczynski et. al].

The spectral Turán problem about graphs with no 6-cycle

A spectral version of extremal graph theory problem is as follows: what is the maximum spectral radius of an H-free graph of order n? Nikiforov posed a conjecture on spectral extremal value of cycles. In the past decade, much attention has been paid to this conjecture and other questions on spectral extremal graphs. In order to approach to Nikiforov's conjecture, we provide a new conjecture on spectral extremal value involving fans and wheels. Moreover, we show it is true for k=2.

Counting and enumerating independent sets with applications to combinatorial optimization problems

Counting and enumerating maximal and maximum independent sets are well-studied problems in graph theory. In this paper we introduce methods to count and enumerate maximal/maximum independent sets in threshold graphs and k-threshold graphs and improve former results for these problems. The results can be applied to combinatorial optimization problems, and in particular to different variations of the knapsack problem. As feasible solutions for instances of those problems correspond to independent sets in threshold graphs and k-threshold graphs, we obtain polynomial time results for special knapsack and multidimensional knapsack instances. Also, we show lower and upper bounds for the number of necessary bins in several bin packing problems.

Removal Lemmas with Polynomial Bounds

Abstract A common theme in many extremal problems in graph theory is the relation between local and global properties of graphs. One of the most celebrated results of this type is the Ruzsa–Szemerédi triangle removal lemma, which states that if a graph is $\varepsilon $-far from being triangle free, then most subsets of vertices of size $C(\varepsilon )$ are not triangle free. Unfortunately, the best known upper bound on $C(\varepsilon )$ is given by a tower-type function, and it is known that $C(\varepsilon )$ is not polynomial in $\varepsilon ^{-1}$. The triangle removal lemma has been extended to many other graph properties, and for some of them the corresponding function $C(\varepsilon )$ is polynomial. This raised the natural question, posed by Goldreich in 2005 and more recently by Alon and Fox, of characterizing the properties for which one can prove removal lemmas with polynomial bounds. Our main results in this paper are new sufficient and necessary criteria for guaranteeing that a graph property admits a removal lemma with a polynomial bound. Although both are simple combinatorial criteria, they imply almost all prior positive and negative results of this type. Moreover, our new sufficient conditions allow us to obtain polynomially bounded removal lemmas for many properties for which the previously known bounds were of tower type. In particular, we show that every semi-algebraic graph property admits a polynomially bounded removal lemma. This confirms a conjecture of Alon.

The partition dimension of subdivision graph on the star

The partition dimension of the graphs is one of the open problems in graph theory. One of the methods which are used researcher is a graph operation, for example, subdivision operations. Let G be a connected graph of order n. The subdivision operation of G is an operation in G that replaces any edge by a path Pki+2 for ki ≥ 1 and i ∈ [1, n]. The result graph of this operation is called the subdivision graph, is denoted by S(G(E; k1, k2, ⋯, kn)). Furthermore, if each ki = 1, then the subdivision graph is denoted by S(G). One of the most recent results on the partition dimension of subdivision operation in the general graphs is published by Amrullah, et al 2016. However, the results show only an upper and lower bounds of the subdivision graph partition dimension. Therefore, this paper is devoted to finding the partition dimension of subdivision graph S(G) and S(G(E; k1, k2, ⋯, kn)) on special star graphs G = K1,n.

No lattice tiling of [formula omitted] by Lee sphere of radius 2

We prove the nonexistence of lattice tilings of Zn by Lee spheres of radius 2 for all dimensions n≥3. This implies that the Golomb-Welch conjecture is true when the common radius of the Lee spheres equals 2 and 2n2+2n+1 is a prime. As a direct consequence, we also answer an open question in the degree-diameter problem of graph theory: the order of any abelian Cayley graph of diameter 2 and degree larger than 5 cannot meet the abelian Cayley Moore bound.

Approximating Partially Bounded Degree Deletion on Directed Graphs

The $\rm{B\small{OUNDED}}$ $\rm{D\small{EGREE}}$ $\rm{D\small{ELETION}}$ problem (BDD) is that of computing a minimum vertex set in a graph $G=(V, E)$ with degree bound $b: V\rightarrow\mathbb{Z}_+$, such that, when it is removed from $G$, the degree of any remaining vertex $v$ is no larger than $b(v)$. It is a classic problem in graph theory and various results have been obtained including an approximation ratio of $2+\ln b_{\max}$ [Okun, Barak, IPL, 2003], where $b_{\max}$ is the maximum degree bound. This paper considers BDD on directed graphs containing unbounded vertices, which we call $\rm{P\small{ARTIAL}}$ $\rm{B\small{OUNDED}}$ $\rm{D\small{EGREE}}$ $\rm{D\small{ELETION}}$ (PBDD). Despite such a natural generalization of standard BDD, it appears that PBDD has never been studied and no algorithmic results are known, approximation or parameterized. It will be shown that 1) in case all the possible degrees are bounded, in-degrees by $b^-$ and out-degrees by $b^+$, BDD on directed graphs can be approximated within $2+\max_{v\in V}\ln(b^-(v)+b^+(v))$, 2) it becomes NP-hard to approximate PBDD better than $b_{\max}$ (even on undirected graphs) once unbounded vertices are allowed, and 3) PBDD can be approximated within $\max\{2,b_{\max}+1\}$ when only in-degrees are partially bounded by $b$ (and the out-degrees of all the vertices are unbounded).

Construction of cospectral graphs

Construction of non-isomorphic cospectral graphs is a nontrivial problem in spectral graph theory especially for large graphs. In this paper, we establish that graph theoretical partial transpose of a graph is a potential tool to create non-isomorphic cospectral graphs by considering a graph as a partitioned graph.

Genetic algorithm-based community detection in large-scale social networks

Communities in social networks are the essential feature which may be considered as a potential parameter in modeling the behavior of the social entities. Detection of communities has attracted a lot of attention in research in social network analysis. It is one of the major challenging problems as it involves high complexity in processing complex web structure. In fact, this problem can be considered as a NP-complete problem in large-scale networks, as this problem is somewhat reducible to the clique problem in graph theory. A number of meta-heuristic algorithms have been proposed to explore the hidden communities. Most of these algorithms have considered the modularity of the network as their objective function. But, the aspect of optimizing the value of modularity is associated with a problem known as resolution limit, where the size of the detected communities depends on the number of edges existing in the network. In this paper, a genetic algorithm-based community detection has been proposed where an efficient single objective function based on similarity matrix has been devised. The similarity index between each pair of nodes has been calculated in a distributed manner over multiple computing nodes. Similarity index proposed in this paper is based on the topological structure of the network. The effectiveness of the proposed approach is examined by comparing the performance with other state-of-the-art community detection algorithms applied over some real-world network datasets.

Linear-time algorithms for finding Hamiltonian and longest [formula omitted]-paths in [formula omitted]-shaped grid graphs

The longest and Hamiltonian path problems are well-known NP-hard problems in graph theory. Despite many applications of these problems, they are still open for many classes of graphs, including solid grid graphs and grid graphs with some holes. We consider the longest and Hamiltonian (s,t)-path problems in C-shaped grid graphs. A (s,t)-path is a path between two given vertices s and t of the graph. A C-shaped grid graph is a rectangular grid graph such that a rectangular grid subgraph is removed from it to make a C-liked shape. In this paper, we first give the necessary conditions for the existence of Hamiltonian cycles and Hamiltonian (s,t)-paths in such graphs. Then by given a linear-time algorithm for finding Hamiltonian cycles and Hamiltonian (s,t)-paths, we show that these necessary conditions are also sufficient. Finally, we give a linear-time algorithm for finding the longest (s,t)-path in these graphs.

Sparsest Feedback Selection for Structurally Cyclic Systems With Dedicated Actuators and Sensors in Polynomial Time

This paper deals with optimal feedback selection problem in linear time-invariant (LTI) structured systems for arbitrary pole placement, an important open problem in structured systems. Specifically, we solve the sparsest feedback selection problem for LTI structured systems. In this paper, we consider structurally cyclic systems with dedicated inputs and outputs. For this class of systems, we prove that the sparsest feedback selection problem is equivalent to the strong connectivity augmentation problem in graph theory. We present an $O(n^2)$ algorithm to find a sparsest feedback matrix for structured systems when every state is actuated by a dedicated input and every state is sensed by a dedicated output, where $n$ denotes the number of states in the system. If the inputs and the outputs are such that not every state is actuated by a dedicated input and/or not every state is sensed by a dedicated output, then we provide an $O(n^3)$ algorithm for the sparsest feedback selection problem. These results show that sparsest feedback selection with dedicated i/o is a specific case of the optimal feedback selection problem that is solvable in polynomial time. We later analyze the sparsest feedback selection problem for structurally cyclic systems when both the input and the output are dedicated and the feedback pattern is constrained . When some of the feedback links are forbidden, we prove that the problem is NP-hard. The results in this paper along with the previously known results conclude that the optimal feedback selection problem is polynomial-time solvable only for the dedicated input–output case without forbidden feedback links and also without weights for the feedback links.

A fuzzy irregular cellular automata-based method for the vertex colouring problem

ABSTRACTVertex colouring is among the most important problems in graph theory which has been widely applied across different real-world problems. In vertex colouring problem (VCP), the goal is to assign a distinct colour to each vertex of the graph in such a way that no two adjacent vertices have the same colour. This paper presents a fuzzy irregular cellular automaton (FICA) for finding a near-optimal solution for the VCP. FICA is an extension fuzzy cellular automaton (FCA) in which the cells of the automaton can be arranged in an irregular structure. The aim of the proposed method is to reap the benefits of both FCA and irregular cellular automata while minimising their drawbacks. To evaluate the proposed method, various computer simulations have been conducted on a variety of graphs. The results suggest that the proposed method is able to achieve better results in terms of the minimum number of required colours and the execution time of the algorithm, compared to other peer algorithms.

Dynamic Service Placement for Virtual Reality Group Gaming on Mobile Edge Cloudlets

To realize mobile virtual reality (VR) group gaming services which are currently hampered by the prohibitive bandwidth and the stringent delay requirements, we investigate the problem of provisioning such services using the emerging mobile edge cloudlet (MEC) networks with a distributed content rendering architecture. The underlying dynamic rendering-module placement problem requires to optimize the service’s operational cost and the users’ end-to-end performance, involving multiple intertwined conflicting system objectives that are discrete, nonconvex, and higher degree polynomial functions with coupled decisions and arbitrary user dynamics over time. We solve this online placement problem by leveraging model predictive control (MPC) and overcoming the aforementioned challenges over each prediction window. We explore the connection between the placement problem and the minimal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> cut problem in graph theory and solve the former via solving a series of instances of the latter. We formally prove the performance guarantee of our approach. We also conduct extensive trace-driven evaluations and demonstrate the superior practical performance of our MPC-based approach compared to the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">de facto</i> practices and the state-of-the-art alternatives.

Constrained ear decompositions in graphs and digraphs

Ear decompositions of graphs are a standard concept related to several major problems in graph theory like the Traveling Salesman Problem. For example, the Hamiltonian Cycle Problem, which is notoriously N P-complete, is equivalent to deciding whether a given graph admits an ear decomposition in which all ears except one are trivial (i.e. of length 1). On the other hand, a famous result of Lovasz states that deciding whether a graph admits an ear decomposition with all ears of odd length can be done in polynomial time. In this paper, we study the complexity of deciding whether a graph admits an ear decomposition with prescribed ear lengths. We prove that deciding whether a graph admits an ear decomposition with all ears of length at most is polynomial-time solvable for all fixed positive integer. On the other hand, deciding whether a graph admits an ear decomposition without ears of length in F is N P-complete for any finite set F of positive integers. We also prove that, for any k ≥ 2, deciding whether a graph admits an ear decomposition with all ears of length 0 mod k is N P-complete. We also consider the directed analogue to ear decomposition, which we call handle decomposition, and prove analogous results : deciding whether a digraph admits a handle decomposition with all handles of length at most is polynomial-time solvable for all positive integer ; deciding whether a digraph admits a handle decomposition without handles of length in F is N P-complete for any finite set F of positive integers (and minimizing the number of handles of length in F is not approximable up to n(1 −)); for any k ≥ 2, deciding whether a digraph admits a handle decomposition with all handles of length 0 mod k is N P-complete. Also, in contrast with the result of Lovasz, we prove that deciding whether a digraph admits a handle decomposition with all handles of odd length is N P-complete. Finally, we conjecture that, for every set A of integers, deciding whether a digraph has a handle decomposition with all handles of length in A is N P-complete, unless there exists h ∈ N such that A = {1, · · · , h}.

Tight Bounds on 1-Harmonious Coloring of Certain Graphs

Graph coloring is one of the most studied problems in graph theory due to its important applications in task scheduling and pattern recognition. The main aim of the problem is to assign colors to the elements of a graph such as vertices and/or edges subject to certain constraints. The 1-harmonious coloring is a kind of vertex coloring such that the color pairs of end vertices of every edge are different only for adjacent edges and the optimal constraint that the least number of colors is to be used. In this paper, we investigate the graphs in which we attain the sharp bound on 1-harmonious coloring. Our investigation consists of a collection of basic graphs like a complete graph, wheel, star, tree, fan, and interconnection networks such as a mesh-derived network, generalized honeycomb network, complete multipartite graph, butterfly, and Benes networks. We also give a systematic and elegant way of coloring for these structures.

Solving Graph Coloring Problem Using an Enhanced Binary Dragonfly Algorithm

The graph coloring problem (GCP) is one of the most interesting classical combinatorial optimization problems in graph theory. It is known to be an NP-Hard problem, so many heuristic algorithms have been employed to solve this problem. In this article, the authors propose a new enhanced binary dragonfly algorithm to solve the graph coloring problem. The binary dragonfly algorithm has been enhanced by introducing two modifications. First, the authors use the Gaussian distribution random selection method for choosing the right value of the inertia weight w used to update the step vector (∆X). Second, the authors adopt chaotic maps to determine the random parameters s, a, c, f, and e. The aim of these modifications is to improve the performance and the efficiency of the binary dragonfly algorithm and ensure the diversity of solutions. The authors consider the well-known DIMACS benchmark graph coloring instances to evaluate the performance of their algorithm. The simulation results reveal the effectiveness and the successfulness of the proposed algorithm in comparison with some well-known algorithms in the literature.

Undergraduate Students’ Solutions of Modeling Problems in Algorithmic Graph Theory

Graphs can be considered as useful mathematical models. Graph algorithms are a common part of undergraduate courses in discrete mathematics. Even though they have been successfully implemented in secondary curricula, little research has been dedicated to the analysis of students’ work. Within a discrete mathematics course for university students, several graph algorithms were introduced via their applications. At the end of the course, the students took a test focused, inter alia, on applications of the algorithms. The mistakes that occurred in 127 students’ solutions of three problems (the Chinese postman problem, the shortest path problem, and the minimum spanning tree problem) were categorized and compared. Surprisingly, no mistakes were identified in the mathematization of situations or in the interpretation of results with respect to the wording of the problem. The categories of errors varied regardless of the problem types. Hierarchical cluster analysis grouped together the students’ solutions for the Chinese postman problem and the minimum spanning tree problem. By means of nonparametric item response theory analysis, the Chinese postman problem was identified as the most problematic for students. Possible sources of this difficulty are discussed in more detail herein.

An Improved Prim Algorithm for Connection Scheme of Last Train in Urban Mass Transit Network

In the network operation stage of urban mass transit, the connection relations between lines can effectively improve travel accessibility. As the last opportunity to travel on the same day, whether the last train can achieve transfer will directly affect passengers’ travel experiences. It is an important work to make the connection scheme of the last train, which needs to take the network structure, the trend, and the volume of passenger flow into overall consideration. This paper analyzes the characters of the last train connection, including the connection structure and the accessible form. Then, we establish a scheme model and define the objectives, the constraints, and the decision data source, and transform it into a graph theory problem. Taking an urban mass transit network as an example, we demonstrate the solution process by using an improved Prim algorithm. Finally, the main aspects and methods of initial scheme optimization are proposed.

Oriented Coloring on Recursively Defined Digraphs

Coloring is one of the most famous problems in graph theory. The coloring problem on undirected graphs has been well studied, whereas there are very few results for coloring problems on directed graphs. An oriented k-coloring of an oriented graph G = ( V , A ) is a partition of the vertex set V into k independent sets such that all the arcs linking two of these subsets have the same direction. The oriented chromatic number of an oriented graph G is the smallest k such that G allows an oriented k-coloring. Deciding whether an acyclic digraph allows an oriented 4-coloring is NP-hard. It follows that finding the chromatic number of an oriented graph is an NP-hard problem, too. This motivates to consider the problem on oriented co-graphs. After giving several characterizations for this graph class, we show a linear time algorithm which computes an optimal oriented coloring for an oriented co-graph. We further prove how the oriented chromatic number can be computed for the disjoint union and order composition from the oriented chromatic number of the involved oriented co-graphs. It turns out that within oriented co-graphs the oriented chromatic number is equal to the length of a longest oriented path plus one. We also show that the graph isomorphism problem on oriented co-graphs can be solved in linear time.

Location Strategy for Traffic Emission Remote Sensing Monitors to Capture the Violated Emissions

Air contamination becomes an urgent problem to be considered as a result of the rapid growth in traffic all over the world. Traffic emissions differ from vehicle to vehicle depending on the vehicle type, production year, fuel octane number, and periodical maintenance of the vehicle. The majority of drivers do not revise their harmful vehicles emissions regularly. Therefore, effective tracking of high-emitting vehicles can be an important solution for reducing traffic air pollution. This study proposes a location strategy for vehicle remote sensing monitors aided with ID-plate recognizer to capture any violated vehicle emissions. The problem is formulated into a graph theory problem, and then a novel adapted metaheuristic algorithm is used to solve the problem. The methodology, using a benchmark problem, has managed to solve the problem to the optimality. Moreover, its robustness is measured statistically.