Graph Labeling Problem Research Articles

On Vertex-Euclidean Deficiency of Complete Fan Graphs and Complete Wheel Graphs

A simple graph G with p vertices is said to be vertex-Euclidean if there exists a bijection f : V ( G ) → { 1 , 2 , … , p } such that f ( v 1 ) + f ( v 2 ) > f ( v 3 ) for each C 3 -subgraph with vertex set { v 1 , v 2 , v 3 } , where f ( v 1 ) < f ( v 2 ) < f ( v 3 ) . More generally, the vertex-Euclidean deficiency of a graph G is the smallest integer k such that G ∪ N k is vertex-Euclidean. To illustrate the idea behind this new graph labeling problem, we study the vertex-Euclidean deficiency of two new families of graphs called the complete fan graphs and the complete wheel graphs. We also explore some related problems, and pose several research topics for further study.

On Vertex-Euclidean Deficiency of Complete Fan Graphs and Complete Wheel Graphs

A simple graph G with p vertices is said to be vertex-Euclidean if there exists a bijection f : V ( G ) → { 1 , 2 , … , p } such that f ( v 1 ) + f ( v 2 ) > f ( v 3 ) for each C 3 -subgraph with vertex set { v 1 , v 2 , v 3 } , where f ( v 1 ) < f ( v 2 ) < f ( v 3 ) . More generally, the vertex-Euclidean deficiency of a graph G is the smallest integer k such that G ∪ N k is vertex-Euclidean. To illustrate the idea behind this new graph labeling problem, we study the vertex-Euclidean deficiency of two new families of graphs called the complete fan graphs and the complete wheel graphs. We also explore some related problems, and pose several research topics for further study.

A graph labeling problem

A graph labeling problem

Population-based iterated greedy algorithm for the S-labeling problem

The iterated greedy metaheuristic generates a sequence of solutions by iterating over a constructive heuristic using destruction and construction phases. In the last few years, it has been employed to solve a considerable number of optimization problems; however, it has never been explored as a solver for graph labeling problems. Hence, in this paper, we contribute to bridging this gap by proposing a population-based iterated greedy to solve the S-labeling problem. The construction phase invokes a novel greedy algorithm for the problem and the destruction phase engages a destructive strength that is attenuated as the algorithm progresses. In addition, it incorporates a restart operator that is activated according to an innovative criterion for detecting convergence. Extensive experiments verify that the proposal can achieve better solution quality than the state-of-the-art optimizer for this optimization problem and other competing algorithms. We have completed the study about the potential of the iterated greedy metaheuristic for graph labeling problems by evaluating experimentally the performance of an extension of the proposed algorithm that solves the Antibandwidth problem. Remarkably, it provides comparable results to those of the best algorithms in the literature for this complex case of labeling problem.

Irregularity and Modular Irregularity Strength of Wheels

It is easily observed that the vertices of a simple graph cannot have pairwise distinct degrees. This means that no simple graph of the order of at least two is, in this way, irregular. However, a multigraph can be irregular. Chartrand et al., in 1988, posed the following problem: in a loopless multigraph, how can one determine the fewest parallel edges required to ensure that all vertices have distinct degrees? This problem is known as the graph labeling problem and, for its solution, Chartrand et al. introduced irregular assignments. The irregularity strength of a graph G is known as the maximal edge label used in an irregular assignment, minimized over all irregular assignments. Thus, the irregularity strength of a simple graph G is equal to the smallest maximum multiplicity of an edge of G in order to create an irregular multigraph from G. In the present paper, we show the existence of a required irregular labeling scheme that proves the exact value of the irregularity strength of wheels. Then, we modify this irregular mapping in six cases and obtain labelings that determine the exact value of the modular irregularity strength of wheels as a natural modification of the irregularity strength.

Distance Two Surjective Labelling of Paths and Interval Graphs

Graph labelling problem has been broadly studied for a long period for its applications, especially in frequency assignment in (mobile) communication system, X -ray crystallography, circuit design, etc. Nowadays, surjective L 2,1 -labelling is a well-studied problem. Motivated from the L 2,1 -labelling problem and the importance of surjective L 2,1 -labelling problem, we consider surjective L 2,1 -labelling ( SL 21 -labelling) problems for paths and interval graphs. For any graph G = V , E , an SL 21 -labelling is a mapping φ : V ⟶ 1,2 , … , n so that, for every pair of nodes u and v , if d u , v = 1 , then φ u − φ v ≥ 2 ; and if d u , v = 2 , then φ u − φ v ≥ 1 , and every label 1,2 , … , n is used exactly once, where d u , v represents the distance between the nodes u and v , and n is the number of nodes of graph G . In the present article, it is proved that any path P n can be surjectively L 2,1 -labelled if n ≥ 4 , and it is also proved that any interval graph IG G having n nodes and degree Δ > 2 can be surjectively L 2,1 -labelled if n = 3 Δ − 1 . Also, we have designed two efficient algorithms for surjective L 2,1 -labelling of paths and interval graphs. The results regarding both paths and interval graphs are the first result for surjective L 2,1 -labelling.

L(2,1,1)-Labeling of Circular-Arc Graphs

Graph labeling problem has been broadly studied in recent past for its wide applications, in mobile communication system for frequency assignment, radar, circuit design, X-ray crystallography, coding theory, etc. An L211-labeling (L211L) of a graph G = (V, E) is a function γ : V → Z∗ such that |γ(u) − γ(v)| ≥ 2, if d(u, v) = 1 and |γ(u) − γ(v)| ≥ 1, if d(u, v) = 1 or 2, where Z∗ be the set of non-negative integers and d(u, v) represents the distance between the nodes u and v. The L211L numbers of a graph G, are denoted by λ2,1,1(G) which is the difference between largest and smallest labels used in L211L. In this article, for circular-arc graph (CAG) G we have proved that λ2,1,1(G) ≤ 6∆ − 4, where ∆ represents the degree of the graph. Beside this we have designed a polynomial time algorithm to label a CAG satisfying the conditions of L211L. The time complexity of the algorithm is O(n∆2), where n is the number of nodes of the graph G.

Radio Mean Labeling Of Paths And Its Total Graph

A graph labeling problem is an assignment of labels to the vertices or edges (or both) of a graph G satisfying some mathematical condition. Radio Mean Labeling, a vertex-labeling of graphs with non-negative integers has a significant application in the study of problems related to radio channel assignment. The maximum label used in a radio mean labeling is called its span, and the lowest possible span of a radio mean labeling is called the radio mean number of a graph. In this paper, we obtain the radio mean number of paths and total graph of paths.

Two-dimensional bandwidth minimization problem: Exact and heuristic approaches

Reducing the bandwidth in a graph is an optimization problem which has generated significant attention over time due to its practical application in Very Large Scale Integration (VLSI) layout designs, solving system of equations, or matrix decomposition, among others. The bandwidth problem is considered as a graph labeling problem where each vertex of the graph receives a unique label. The target consists in finding an embedding of the graph in a line, according to the labels assigned to each vertex, that minimizes the maximum distance between the labels of adjacent vertices. In this work, we are focused on a 2D variant where the graph has to be embedded in a two-dimensional grid instead. To solve it, we have designed two constructive and three local search methods which are integrated in a Basic Variable Neighborhood Search (BVNS) scheme. To assess their performance, we have designed three Constraint Satisfaction Problem (CSP) models. The experimental results show that our CSP models obtain remarkable results in small or medium size instances. On the other hand, BVNS is capable of reaching equal or similar results than the CSP models in a reduced run-time for small instances, and it can provide high quality solutions in those instances which are not optimally solvable with our CSP models.

An Injective Version of the 1-2-3 Conjecture

In this work, we introduce and study a new graph labelling problem standing as a combination of the 1-2-3 Conjecture and injective colouring of graphs, which also finds connections with the notion of graph irregularity. In this problem, the goal, given a graph G, is to label the edges of G so that every two vertices having a common neighbour get incident to different sums of labels. We are interested in the minimum k such that G admits such a k-labelling. We suspect that almost all graphs G can be labelled this way using labels $$1,\dots ,\Delta (G)$$ . Towards this speculation, we provide bounds on the maximum label value needed in general. In particular, we prove that using labels $$1,\dots ,\Delta (G)$$ is indeed sufficient when G is a tree, a particular cactus, or when its injective chromatic number $$\mathrm{\chi _{\mathrm{i}}}(G)$$ is equal to $$\Delta (G)$$ .

A new iterated local search algorithm for the cyclic bandwidth problem

The Cyclic Bandwidth Problem is an important graph labeling problem with numerous applications. This work aims to advance the state-of-the-art of practically solving this computationally challenging problem. We present an effective heuristic algorithm based on the general iterated local search framework and integrating dedicated search components. Specifically, the algorithm relies on a simple, yet powerful local optimization procedure reinforced by two complementary perturbation strategies. The local optimization procedure discovers high-quality solutions in a particular search zone while the perturbation strategies help the search to escape local optimum traps and explore unvisited areas. We present intensive computational results on 113 benchmark instances from 8 different families, and show performances that are never achieved by current best algorithms in the literature.

Accurate Long-Term Multiple People Tracking using Video and Body-Worn IMUs.

Most modern approaches for video-based multiple people tracking rely on human appearance to exploit similarities between person detections. Consequently, tracking accuracy degrades if this kind of information is not discriminative or if people change apparel. In contrast, we present a method to fuse video information with additional motion signals from body-worn inertial measurement units (IMUs). In particular, we propose a neural network to relate person detections with IMU orientations, and formulate a graph labeling problem to obtain a tracking solution that is globally consistent with the video and inertial recordings. The fusion of visual and inertial cues provides several advantages. The association of detection boxes in the video and IMU devices is based on motion, which is independent of a person's outward appearance. Furthermore, inertial sensors provide motion information irrespective of visual occlusions. Hence, once detections in the video are associated with an IMU device, intermediate positions can be reconstructed from corresponding inertial sensor data, which would be unstable using video only. Since no dataset exists for this new setting, we release a dataset of challenging tracking sequences, containing video and IMU recordings together with ground-truth annotations. We evaluate our approach on our new dataset, achieving an average IDF1 score of 91.2%. The proposed method is applicable to any situation that allows one to equip people with inertial sensors.

Algorithmic expedients for the S-labeling problem

Graph labeling problems have been widely studied in the last decades and have a vast area of application. In this work, we study the recently introduced S-labeling problem, in which the nodes get labeled using labels from 1 to |V| and for each edge the contribution to the objective function, called S-labeling number of the graph, is the minimum label of its end-nodes. The goal is to find a labeling with minimum value. The problem is NP-hard for planar subcubic graphs, although for many other graph classes the complexity status is still unknown.In this paper, we present different algorithmic approaches for tackling this problem: We develop an exact solution framework based on Mixed-Integer Programming (MIP) which is enhanced with valid inequalities, starting and primal heuristics and specialized branching rules. We show that our MIP formulation has no integrality gap for paths, cycles and perfect n-ary trees, and, to the best of our knowledge, we give the first polynomial-time algorithm for the problem on n-ary trees as well as a closed formula for the S-labeling number for such trees. Moreover, we also present a Lagrangian heuristic and a constraint programming approach. A computational study is carried out in order to (i) investigate if there may be other special graph classes, where our MIP formulation has no integrality gap, and (ii) assess the effectiveness of the proposed solution approaches for solving the problem on a dataset consisting of general graphs.

Selecting subexpressions to materialize at datacenter scale

We observe significant overlaps in the computations performed by user jobs in modern shared analytics clusters. Naïvely computing the same subexpressions multiple times results in wasting cluster resources and longer execution times. Given that these shared cluster workloads consist of tens of thousands of jobs, identifying overlapping computations across jobs is of great interest to both cluster operators and users. Nevertheless, existing approaches support orders of magnitude smaller workloads or employ heuristics with limited effectiveness. In this paper, we focus on the problem of subexpression selection for large workloads, i.e., selecting common parts of job plans and materializing them to speed-up the evaluation of subsequent jobs. We provide an ILP-based formulation of our problem and map it to a bipartite graph labeling problem. Then, we introduce B ig S ubs , a vertex-centric graph algorithm to iteratively choose in parallel which subexpressions to materialize and which subexpressions to use for evaluating each job. We provide a distributed implementation of our approach using our internal SQL-like execution framework, SCOPE, and assess its effectiveness over production workloads. B ig S ubs supports workloads with tens of thousands of jobs, yielding savings of up to 40% in machine-hours. We are currently integrating our techniques with the SCOPE runtime in our production clusters.

Constraint programming models and population-based simulated annealing algorithm for finding graceful and $$\alpha $$ α -labeling of quadratic graphs

In this research, a new mathematical integer programming model is presented for the graph labeling problem of quadratic graphs. The advantages of this model are linearity and the existence of an objective function. Furthermore, two constraint programming models and a meta-heuristics algorithm are also developed to generate feasible graceful labeling and $$\alpha $$ -labeling for special classes of quadratic graphs. Experimental results on large sizes of graphs from the literature show the efficiency of the proposed model and approach.

Scheduling Resource-Bounded Monitoring Devices for Event Detection and Isolation in Networks

In networked systems, monitoring devices such as sensors are typically deployed to monitor various target locations. Targets are the points in the physical space at which events of some interest, such as random faults or attacks, can occur. Most often, monitoring devices have limited energy supplies, and they can operate for a limited duration. As a result, energy-efficient monitoring of target locations through a set of monitoring devices with limited energy supplies is a crucial problem in networked systems. In this paper, we study optimal scheduling of monitoring devices to maximize network coverage for detecting and isolating events on targets for a given network lifetime. The monitoring devices considered could remain active only for a fraction of the overall network lifetime. We formulate the problem of scheduling of monitoring devices as a graph labeling problem, which unlike other existing solutions, allows us to directly utilize the underlying network structure to explore the trade-off between coverage and network lifetime. In this direction, first we present a greedy heuristic, and then a game-theoretic solution to the graph labeling problem. The proposed setup can be used to simultaneously solve the scheduling and placement of monitoring devices, which, as our simulations illustrate, gives improved performance as compared to separately solving the placement and scheduling problems. Finally, we illustrate our results on various networks, including real-world water distribution networks and random geometric networks.

Diophantine Edge Graceful Graph

Background: Graph labeling problems have interesting applications in coding theory, communication networks, optimal circuits’ layouts and graph decomposition problems, as described in various patents. Keywords: Diophantine edge graceful, complete (m, h) trees, corona of graphs.

Italian domination in trees

The Roman domination number and Italian domination number (also known as the Roman {2}-domination number) are graph labeling problems in which each vertex is labeled with either 0, 1, or 2. In the Roman domination problem, each vertex labeled 0 must be adjacent to at least one vertex labeled 2. In the Italian domination problem, each vertex labeled 0 must have the labels of the vertices in its closed neighborhood sum to at least two. The Italian domination number, γI(G), of a graph G is the minimum possible sum of such a labeling, where the sum is taken over all the vertices in G. It is known that if T is a tree with at least two vertices, then γ(T)+1≤γI(T)≤2γ(T). In this paper, we characterize the trees T for which γ(T)+1=γI(T), and we characterize the trees T for which γI(T)=2γ(T).

Relabelling vertices according to the network structure by minimizing the cyclic bandwidth sum

Getting a labelling of vertices close to the structure of the graph has been proved to be of interest in many applications, e.g. to follow signals indexed by the vertices of the network. This question can be related to a graph labelling problem known as the cyclic bandwidth sum problem (CBSP). It consists of finding a labelling of the vertices of an undirected and unweighted graph with distinct integers such that the sum of (cyclic) difference of labels of adjacent vertices is minimized. In this paper, we introduce a new heuristic to follow the structure of the graph, by finding an approximate solution for the CBSP. Although theoretical results exist that give optimal value of cyclic bandwidth sum (CBS) for standard graphs, there are neither results for real-world complex networks, nor explicit methods to reach this optimal result. Furthermore, only a few methods have been proposed to approximately solve this problem. The heuristic we propose is a two-step algorithm: the first step consists of traversing the graph to find a set of paths which follow the structure of the graph, using a similarity criterion based on the Jaccard index to jump from one vertex to the next one. The second step is the merging of all obtained paths, based on a greedy approach that extends a partial solution by inserting a new path at the position that minimizes the CBS. The effectiveness of the proposed heuristic is shown through experiments on graphs whose optimal value of CBS is known, as well as on complex networks, where the consistency between labelling and topology is highlighted.

WrapIt

Wire wrapping is a traditional form of handmade jewelry that involves bending metal wire to create intricate shapes. The technique appeals to novices and casual crafters because of its low cost, accessibility and unique aesthetic. We present a computational design tool that addresses the two main challenges of creating 2D wire-wrapped jewelry: decomposing an input drawing into a set of wires, and bending the wires to give them shape. Our main contribution is an automatic wire decomposition algorithm that segments a drawing into a small number of wires based on aesthetic and fabrication principles. We formulate the task as a constrained graph labeling problem and present a stochastic optimization approach that produces good results for a variety of inputs. Given a decomposition, our system generates a 3D-printed custom support structure, or jig , that helps users bend the wire into the appropriate shape. We validated our wire decomposition algorithm against existing wire-wrapped designs, and used our end-to-end system to create new jewelry from clipart drawings. We also evaluated our approach with novice users, who were able to create various pieces of jewelry in less than half an hour.