Edge Set Research Articles

Flip paths between lattice triangulations

We present a O(n32)-time algorithm for the shortest (diagonal) flip path problem for lattice triangulations with n points, improving over previous O(n2)-time algorithms. For a large, natural class of inputs, our bound is tight in the sense that our algorithm runs in time linear in the number of flips in the output flip path. Our results rely on an independently interesting structural elucidation of shortest flip paths as the linear orderings of a unique partially ordered set, called a minimum flip plan, constructed by a novel use of Farey sequences from elementary number theory. Flip paths between general (not necessarily lattice) triangulations have been studied in the combinatorial setting for nearly a century. In the Euclidean geometric setting, finding a shortest flip path between two triangulations is NP-complete. However, for lattice triangulations, which are studied as spin systems, there are known On2-time algorithms to find shortest flip paths. These algorithms, as well as ours, apply to constrained flip paths that ensure a set of constraint edges are present in every triangulation along the path. Implications for determining simultaneously flippable edges, i.e. finding optimal simultaneous flip paths between lattice triangulations, and for counting lattice triangulations are discussed.

Probabilistic Analysis of Optimization Problems on Sparse Random Shortest Path Metrics

Simple heuristics for (combinatorial) optimization problems often show a remarkable performance in practice. Worst-case analysis often falls short of explaining this performance. Because of this, “beyond worst-case analysis” of algorithms has recently gained a lot of attention, including probabilistic analysis of algorithms. The instances of many (combinatorial) optimization problems are essentially a discrete metric space. Probabilistic analysis for such metric optimization problems has nevertheless mostly been conducted on instances drawn from Euclidean space, which provides a structure that is usually heavily exploited in the analysis. However, most instances from practice are not Euclidean. Little work has been done on metric instances drawn from other, more realistic, distributions. Some initial results have been obtained in recent years, where random shortest path metrics generated from dense graphs (either complete graphs or Erdős–Rényi random graphs) have been used so far. In this paper we extend these findings to sparse graphs, with a focus on sparse graphs with ‘fast growing cut sizes’, i.e. graphs for which |delta (U)|=Omega (|U|^varepsilon ) for some constant varepsilon in (0,1) for all subsets U of the vertices, where delta (U) is the set of edges connecting U to the remaining vertices. A random shortest path metric is constructed by drawing independent random edge weights for each edge in the graph and setting the distance between every pair of vertices to the length of a shortest path between them with respect to the drawn weights. For such instances generated from a sparse graph with fast growing cut sizes, we prove that the greedy heuristic for the minimum distance maximum matching problem, and the nearest neighbor and insertion heuristics for the traveling salesman problem all achieve a constant expected approximation ratio. Additionally, for instances generated from an arbitrary sparse graph, we show that the 2-opt heuristic for the traveling salesman problem also achieves a constant expected approximation ratio.

QSPR analysis through graph models for predicting ADMET properties of antifungal drugs to treat fungal diseases

AbstractThe chemical structure of a drug is considered as a chemical graph where the vertex set is the set of atoms and the edge set is the set of bonds between the atoms. Fungi‐related diseases are becoming a more serious medical problem as a result of changes in the worldwide environment. In this article, the QSPR analysis is performed to predict the ADMET properties of the drugs used to treat fungal infections such as mucormycosis, blastomycosis, invasive candidiasis, talaromycosis, cryptococcus, neoformans, and so forth. The antifungal drugs posaconazole, isavuconazole, and amphotericin B and its analogs are considered to carry out the QSPR analysis. The domination numbers of these drugs are determined to correlate with their ADMET properties through cubic regression and the analysis suggests a high association between the domination numbers of drugs and their ADMET properties. The ADMET properties of the analog structures of the drug amphotericin B are also predicted in this QSPR analysis.

Graph embedding based multi-label Zero-shot Learning

Multi-label Zero-shot Learning (ZSL) is more reasonable and realistic than standard single-label ZSL because several objects can co-exist in a natural image in real scenarios. Intra-class feature entanglement is a significant factor influencing the alignment of visual and semantic features, resulting in the model’s inability to recognize unseen samples comprehensively and completely. We observe that existing multi-label ZSL methods place a greater emphasis on attention-based refinement and decoupling of visual features, while ignoring the relationship between label semantics. Relying on label correlations to solve multi-label ZSL tasks has not been deeply studied. In this paper, we make full use of the co-occurrence relationship between category labels and build a directed weighted semantic graph based on statistics and prior knowledge, in which node features represent category semantics and weighted edges represent conditional probabilities of label co-occurrence. To guide the targeted extraction of visual features, node features and edge set weights are simultaneously updated and refined, and embedded into the visual feature extraction network from a global and local perspective. The proposed method’s effectiveness was demonstrated by simulation results on two challenging multi-label ZSL benchmarks: NUS-WIDE and Open Images. In comparison to state-of-the-art models, our model achieves an absolute gain of 2.4% mAP on NUS-WIDE and 2.1% mAP on Open Images respectively.

Patterns and Line-Adjacency Matrix Strategy on a Game of Induced Matching

A matching is an independent set of edges in a graph G; an induced matching is a matching with an additional property that no two of its edges are joined by an edge in G. An induced matching M in a graph G is maximal if no other induced matching in G contains M. In the proposed game, players alternate choose edges on a graph while maintaining an induced matching. They continue until the matching is maximal (i.e. no player can choose another edge), and the last player to choose an edge wins. This paper will discuss patterns for this game and prove results on several categories of graphs including Complete Graphs, Path Graphs, Cycle Graphs, and Ladder Graphs. In addition, a novel Line-Adjacency Matrix method has been defined and proved to calculate all possible edges for the next move in a connected graph. Keywords: Induced Matching, Adjacency Matrix, Line-Adjacency Graph

On algorithmic complexity of imprecise spanners

Let t>1 be a real number. A geometric t-spanner is a geometric graph for a point set in Rd with straight line segments between vertices such that the ratio of the shortest-path distance between every pair of vertices in the graph (with Euclidean edge lengths) to their actual Euclidean distance is at most t.An imprecise point set is modeled by a set R of regions in Rd. If one chooses a point inside each region of R, then the resulting point set is called a precise instance from R. An imprecise t-spanner for an imprecise point set R is a graph G=(R,E) such that for each precise instance S from R, graph GS=(S,ES), where ES is the set of edges corresponding to E and S, is a t-spanner.In this paper, we show an imprecise point set R of n straight-line segments in the plane such that any imprecise t-spanner for R has Ω(n2) edges. Then, we give an algorithm that computes an imprecise t-spanner for a set of n pairwise disjoint d-dimensional balls with arbitrary sizes. This imprecise t-spanner has O(n/(t−1)d) edges and can be computed in O(nlog⁡n/(t−1)d) time. Finally, we show that given an imprecise spanner, finding a precise instance such that its corresponding precise spanner has minimum dilation between all possible precise instances of the imprecise spanner is NP-hard, no matter if crossing edges are allowed or not.

Independent k-rainbow bondage number of graphs

For an integer an independent k-rainbow dominating function (IkRDF for short) on a graph G is a function g that assigns to each vertex a set of colors chosen from the subsets of satisfying the following conditions: (i) if , then , and (ii) the set is an independent set. The weight of an IkRDF g is the value . The independent k-rainbow domination number is the minimum weight of an IkRDF on G. In this paper, we initiate a study of the independent k-rainbow bondage number of a graph G having at least one component of order at least three, defined as the smallest size of set of edges for which . We begin by showing that the decision problem associated with the independent k-rainbow bondage problem is NP-hard for general graphs for . Then various upper bounds on are established as well as exact values on it for some special graphs. In particular, for trees T of order at least three, it is shown that .

Weak saturation stability

The paper studies wsat(G,H) which is the minimum number of edges in a weakly H-saturated subgraph of G. We prove that wsat(Kn,H) is ‘stable’ – remains the same after independent removal of every edge of Kn with constant probability – for all pattern graphs H such that there exists a ‘local’ set of edges percolating in Kn. This is true, for example, for cliques and complete bipartite graphs. We also find a threshold probability for the weak K1,t-saturation stability.

On the complexity of the storyplan problem

We study the problem of representing a graph as a storyplan, a recently introduced model for dynamic graph visualization. It is based on a sequence of frames, each showing a subset of vertices and a planar drawing of their induced subgraphs, where vertices appear and disappear over time. Namely, in the StoryPlan problem, we are given a graph and we want to decide whether there exists a total vertex appearance order for which a storyplan exists. We prove that the problem is NP-complete, and complement this hardness with two parameterized algorithms, one in the vertex cover number and one in the feedback edge set number of the input graph. We prove that partial 3-trees always admit a storyplan, which can be computed in linear time. Finally, we show that the problem remains NP-complete if the vertex appearance order is given and we have to choose how to draw the frames.

Permutation graphs with co-inversion on Γ<sub>1</sub> - non-deranged permutations

In this paper, we define permutation graphs on Γ1 -non-deranged permutations using the set of co-inversion as edge set, and the values of permutation as the set of vertices. From the graphs, we observed that diameter and radius of the graph of any ω1 is one, the graph of any ωp-1 ∈ GΓ1p simple, the graph of ω1 is completed and other properties of the graphs were also observed.

The perfect matching and tight Hamilton cycle decomposition of complete n-balanced mk-partite k-uniform hypergraphs

Let r≥k≥2 and Kr×n(k) denote the complete n-balanced r-partite k-uniform hypergraph, whose vertex set consists of r parts, each part has n vertices, and whose edge set contains all the k-element subsets with no two vertices from one part. A Hamilton cycle decomposition of Kr×n(k) is a partition of the edge set E(Kr×n(k)) into Hamilton cycles. In this paper, we consider the perfect matching decomposition and the tight Hamilton cycle decomposition of Kmk×n(k) for m≥2. We obtain the following results. (1) Let k≥3, m≥1 and n≥1. Then Kmk×n(k) has a perfect matching decomposition. (2) If Kmk(k) has a Hamilton cycle decomposition, then Kmk×n(k) has a tight Hamilton cycle decomposition.

On the edge dimension and the fractional edge dimension of graphs

Let G be a graph with vertex set V ( G ) and edge set E ( G ) , and let d ( u , w ) denote the length of a u − w geodesic in G . For any vertex v ∈ V ( G ) and any edge e = x y ∈ E ( G ) , let d ( e , v ) = min { d ( x , v ) , d ( y , v ) } . For any distinct edges e 1 , e 2 ∈ E ( G ) , let R { e 1 , e 2 } = { z ∈ V ( G ) : d ( z , e 1 ) ≠ d ( z , e 2 ) } . Kelenc, Tratnik and Yero [Discrete Appl. Math. 251 (2018) 204-220] introduced the notion of an edge resolving set and the edge dimension of a graph: A vertex subset S ⊆ V ( G ) is an edge resolving set of G if | S ∩ R { e 1 , e 2 } | ≥ 1 for any distinct edges e 1 , e 2 ∈ E ( G ) , and the edge dimension , edim ( G ) , of G is the minimum cardinality among all edge resolving sets of G . For a function g defined on V ( G ) and for U ⊆ V ( G ) , let g ( U ) = ∑ s ∈ U g ( s ) . A real-valued function g : V ( G ) → [ 0 , 1 ] is an edge resolving function of G if g ( R { e 1 , e 2 } ) ≥ 1 for any distinct edges e 1 , e 2 ∈ E ( G ) . The fractional edge dimension , edim f ( G ) , of G is min { g ( V ( G ) ) : g is an edge resolving function of G } . Note that edim f ( G ) reduces to edim ( G ) if the codomain of edge resolving functions is restricted to { 0 , 1 } . In this paper, we introduce and study the fractional edge dimension of graphs, and we obtain some general results on the edge dimension of graphs. We show that there exist two non-isomorphic graphs on the same vertex set with the same edge metric coordinates. We construct two graphs G and H such that H ⊂ G and both edim ( H ) − edim ( G ) and edim f ( H ) − edim f ( G ) can be arbitrarily large. We show that a graph G with edim ( G ) = 2 cannot have K 5 or K 3 , 3 as a subgraph, and we construct a non-planar graph H satisfying edim ( H ) = 2 . It is easy to see that, for any connected graph G of order at least three, 1 ≤ edim f ( G ) ≤ | V ( G ) | 2 ; we characterize graphs G satisfying edim f ( G ) = 1 and examine some graph classes satisfying edim f ( G ) = | V ( G ) | 2 . We also determine the fractional edge dimension for some classes of graphs.

Eccentric connectivity index in transformation graph Gxy+

Abstract Let G be a connected graph with vertex set V(G)and edge set E(G). The eccentric connectivity index of G is defined as ∑ ν ∈ V ( G ) ec ( ν ) deg ( ν ) \sum\limits_{\nu\in{\rm{V}}\left({\rm{G}}\right)}{{\rm{ec}}\left(\nu\right)\,{\rm{deg}}\left(\nu\right)} where ec(v) the eccentricity of a vertex v and deg(v)is its degree and denoted by ɛc(G). In this paper, we investigate the eccentric connectivity index of the transformation graph Gxy+.

Tool wear area estimation through in-process edge force coefficient in trochoidal milling of Inconel 718

The rapid wear and premature failure of the cutting tool are prone to happen due to increased forces during machining difficult-to-cut materials such as Inconel 718. The application of alternative toolpath such as trochoidal milling has significantly improved tool life and reduced the overall cycle time of the process. The wear pattern of the tool has a direct impact on the cutting forces, which increases with tool deterioration. The cutting forces in milling are modeled through the mechanistic force model and can be designated through a set of force coefficients, i.e. cutting and edge representing the shearing and ploughing phenomenon of metal removal. It has been established in the literature that tool wear has a considerable effect on the value of edge force coefficients. This paper aims to determine the in-process edge force coefficients for the trochoidal toolpath and correlates them with the corresponding flank wear area. The proposed correlation will further assist in predicting the level of flank wear area based on the force values in trochoidal milling.

Maximal matching cover pebbling number for variants of hypercube

An edge pebbling move is defined as the removal of two pebbles from one edge and placing one on the adjacent edge. The maximal matching cover pebbling number, fmmcp(G), of a graph G, is the minimum number of pebbles that must be placed on E(G), such that after a sequence of pebbling moves the set of edges with pebbles forms a maximal matching regardless of the initial configuration. In this paper, we find the maximal matching cover pebbling number for variants of hypercube.

Extreme and statistical properties of eigenvalue indices of simple connected graphs

We analyze graphs attaining the extreme values of various spectral indices in the class of all simple connected graphs, as well as in the class of graphs which are not complete multipartite graphs. We also present results on density of spectral gap indices and its nonpersistency with respect to small perturbations of the underlying graph. We show that a small change in the set set of edges may result in a significant change of the spectral index like, e.g., the spectral gap or spectral index. We also present a statistical and numerical analysis of spectral indices of graphs of the order m≤10. We analyze the extreme values for spectral indices for graphs and their small perturbations. Finally, we present the statistical and extreme properties of graphs on m≤10 vertices.

A Self-Adaptive Evolutionary Deception Framework for Community Structure

The rapid development of community detection algorithms, while serving users in social networks, also brings about certain privacy problems. In this work, we study community deception, which aims to counter malicious community detection attacks by imperceptibly modifying a small part of the connections. However, it is computationally challenging to find an optimal edge set since it is an NP-hard problem. To address this issue, we propose a self-adaptive evolutionary deception (SAEP) framework. In SAEP, a novel fitness function that is able to capture local and global community change is being proposed. SAEP also provides a well-designed initialization mechanism to reduce the size of the solution space. In addition, we assign an indicator to each gene to reflect its strength within the chromosome that it belongs to, thereby a set of self-adaptive operations can be defined to enhance the algorithm’s stability and efficacy. Furthermore, we define a new “edge distance” to conserve the limited modification resource on the graph. In the experiment, the proposed method is tested against different community detection methods using various real-world datasets, and the experimental results demonstrate that SAEP improves significantly over state-of-the-art approaches in terms of effectiveness.

Another Look at Geodetic Hop Domination in a Graph

Let $G$ be an undirected graph with vertex and edge sets $V(G)$ and $E(G)$, respectively. A subset $S$ of vertices of $G$ is a geodetic hop dominating set if it is both a geodetic and a hop dominating set. The geodetic hop domination number of $G$ is the minimum cardinality among all geodetic hop dominating sets in $G$. Geodetic hop dominating sets in a graph resulting from the join of two graphs have been characterized. These characterizations have been used to determine the geodetic hop domination number of the graphs considered. A realization result involving the hop domination number and geodetic hop domination number is also obtained.

Extension of some edge graph problems: Standard, parameterized and approximation complexity

We consider extension variants of some edge optimization problems in graphs containing the classical Edge Cover, Matching, and Edge Dominating Set problems and generalizations thereof. Given a graph G=(V,E) and an edge set U⊆E, it is asked whether there exists an inclusion-wise minimal (or maximal, respectively) feasible solution E′ which satisfies a given property, for instance, being an edge dominating set (or a matching, respectively) and containing the forced edge set U (or avoiding any edges from the forbidden edge set E∖U, respectively). We present hardness results for these problems, for restricted instances such as bipartite or planar graphs. We counter-balance these negative results with parameterized complexity results. We also consider the price of extension, a natural optimization problem variant of extension problems, leading to some approximation and inapproximability results.

Local Metric Dimension of Certain Classes of Circulant Networks

Let G(V,E) be a graph with a set of vertices V and a set of edges E. Then, a minimum subset Wl of V is said to be a local metric basis of G if for any two adjacent vertices u,v∈V∖Wl there exists a vertex w∈Wl such that d(u,w) ≠ d(v,w). The cardinality of a local metric basis is referred to as the local metric dimension of the graph G denoted by βl(G). In this paper, we investigate the local metric dimensions of certain circulant-related architectures such as Harary graphs Hk,n with even k or n, Toeplitz networks, and ILLIAC networks.