Hull Set Research Articles

On the hull number on cycle convexity of graphs

In this work, we study the parameter hull number in a recently defined graph convexity called Cycle Convexity, whose definition is motivated by related notions in Knot Theory.For a graph G=(V,E), define the interval function in the Cycle Convexity as Icc(S)=S∪{v∈V(G)|there is a cycle C in G such that V(C)∖S={v}}, for every S⊆V(G). We say that S⊆V(G) is convex if Icc(S)=S. The convex hull of S⊆V(G), denoted by Hull(S), is the inclusion-wise minimal convex set S′ such that S⊆S′. A set S⊆V(G) is called a hull set if Hull(S)=V(G). The hull number of G in the cycle convexity, denoted by hncc(G), is the cardinality of a smallest hull set of G.We first focus on the class of planar graphs, as the main motivation for the definition of hncc(G) stems from Knot Theory and occurs when G is a 4-regular planar graph. We prove that: the hull number of a 4-regular planar graph is at most half of its number of vertices and that such bound is tight; and that deciding whether hncc(G)≤k, provided a positive integer k and a planar graph G, is an NP-complete problem.On the positive side, we present polynomial-time algorithms to compute the hull number in the cycle convexity of chordal graphs, P4-sparse graphs, and grids.

Computational and structural aspects of the geodetic and the hull numbers of shadow graphs

Given a set X ⊆ V ( G ) , let [ X ] G denote the set of all vertices of G lying on some shortest path between two vertices of X . If [ X ] G = X , then X is a convex set of G . The convex hull of X , denoted by 〈 X 〉 G , is the smallest convex set containing X . We say that X is a hull set of G if 〈 X 〉 G = V ( G ) and that X is a geodetic set of G if [ X ] G = V ( G ) . The hull number h ( G ) of G is the minimum cardinality of a hull set of G ; and the geodetic number g ( G ) of G is the minimum cardinality of a geodetic set of G . The shadow graph of G , denoted by S ( G ) , is the graph obtained from G by adding a new vertex v ′ for each vertex v of G and joining v ′ to the neighbors of v in G . In this paper, we present sharp bounds for the geodetic and the hull numbers of shadow graphs. We characterize the classes of graphs for which some of these bounds are attained. We also prove for a fixed integer k , that the problem of deciding whether g ( S ( G ) ) ≤ n − k is NP -complete even if the diameter of G is 2; and that the problem of deciding whether h ( S ( G ) ) ≤ n − 1 belongs to P .

The hull number in the convexity of induced paths of order 3

A set S of vertices of a graph G is P3⁎-convex if there is no vertex outside S having two non-adjacent neighbors in S. The P3⁎-convex hull of S is the minimum P3⁎-convex set containing S. If the P3⁎-convex hull of S is V(G), then S is a P3⁎-hull set. The minimum size of a P3⁎-hull set is the P3⁎-hull number of G. In this paper, we show that the problem of deciding whether the P3⁎-hull number of a chordal graph is at most k is NP-complete and present a linear-time algorithm to determine this parameter and provide a minimum P3⁎-hull set for unit interval graphs.

Computational and structural aspects of the geodetic and the hull numbers of shadow graphs

Given a set X⊆V(G), let [X]G denote the set of all vertices of G lying on some shortest path between two vertices of X. If [X]G=X, then X is a convex set of G. The convex hull of X, denoted by 〈X〉G, is the smallest convex set containing X. We say that X is a hull set of G if 〈X〉G=V(G) and that X is a geodetic set of G if [X]G=V(G). The hull number h(G) of G is the minimum cardinality of a hull set of G; and the geodetic number g(G) of G is the minimum cardinality of a geodetic set of G. The shadow graph of G, denoted by S(G), is the graph obtained from G by adding a new vertex v′ for each vertex v of G and joining v′ to the neighbors of v in G. In this paper, we present sharp bounds for the geodetic and the hull numbers of shadow graphs. We characterize the classes of graphs for which some of these bounds are attained. We also prove for a fixed integer k, that the problem of deciding whether g(S(G))≤n−k is NP-complete even if the diameter of G is 2; and that the problem of deciding whether h(S(G))≤n−1 belongs to P.

Hull and geodetic numbers for some classes of oriented graphs

Let D be an orientation of a simple graph. Given u , v ∈ V ( D ) , a directed shortest ( u , v ) -path is a ( u , v ) -geodesic. S ⊆ V ( D ) is convex if, for every u , v ∈ S , the vertices in each ( u , v ) -geodesic and in each ( v , u ) -geodesic are in S . For each S ⊆ V ( D ) the (convex) hull of S , denoted by [ S ] , is the smallest convex set containing S . S ⊆ V ( D ) is a hull set if [ S ] = V ( D ) . S ⊆ V ( D ) is a geodetic set of D if each vertex of D lies in a ( u , v ) -geodesic, for some u , v ∈ S . The cardinality of a minimum hull set (resp. geodetic set) of G is the hull number (resp. geodetic number) of D , denoted by hn ⃗ ( D ) (resp. gn ⃗ ( D ) ). We first show a tight upper bound on hn ⃗ ( D ) . Given k ∈ Z + ∗ , we prove that deciding if hn ⃗ ( D ) ≤ k is NP -complete when D is an oriented partial cube; and if gn ⃗ ( D ) ≤ k is W [2]-hard parameterized by k and has no ( c ⋅ ln n ) -approximation algorithm, unless P = NP , even if D has an underlying graph that is bipartite or split or cobipartite. We also show polynomial-time algorithms to compute hn ⃗ ( D ) and gn ⃗ ( D ) when D is an oriented cactus.

Robust control for incremental quadratic constrained nonlinear time-delay systems subject to actuator saturation

This article presents a saturation controller design method for a type of nonlinear time-delay systems. Different from the current works, the nonlinear systems considered herein satisfy an incremental quadratic constraint, which is a more general nonlinearity. Firstly, for the constraint in the actuator, the convex set theory is employed to transform the saturated feedback law into a parameterized linear convex hull set. Next, the convex hull Lyapunov function is constructed to obtain sufficient conditions for exponential stability of the closed-loop system. In addition, by solving an optimization problem, the maximal estimation of domain of attraction is also determined. Finally, numerical examples are provided to verify the proposed method.

On the geodetic hull number for complementary prisms II

In the geodetic convexity, a set of vertices S of a graph G is convex if all vertices belonging to any shortest path between two vertices of S lie in S. The convex hull H(S) of S is the smallest convex set containing S. If H(S) = V (G), then S is a hull set. The cardinality h(G) of a minimum hull set of G is the hull number of G. The complementary prism GḠ of a graph G arises from the disjoint union of the graph G and Ḡ by adding the edges of a perfect matching between the corresponding vertices of G and Ḡ. A graph G is autoconnected if both G and Ḡ are connected. Motivated by previous work, we study the hull number for complementary prisms of autoconnected graphs. When G is a split graph, we present lower and upper bounds showing that the hull number is unlimited. In the other case, when G is a non-split graph, it is limited by 3.

Hull and Geodetic Numbers for Some Classes of Oriented Graphs

An oriented graph D is an orientation of a simple graph, i.e. a directed graph whose underlying graph is simple. A directed path from u to v with minimum number of arcs in D is an (u, v)-geodesic, for every u, v ∈ V(D). A set S ⊆ V(D) is (geodesically) convex if, for every u, v ∈ S, all the vertices in each (u, v)-geodesic and in each (v, u)-geodesic are in S. For every S ⊆ V(D) the (convex) hull of S is the smallest convex set containing S and it is denoted by [S]. A hull set of D is a set S ⊆ V(D) whose hull is V(D). The cardinality of a minimum hull set is the hull number of D and it is denoted by hn→(D). A geodetic set of D is a set S ⊆ V(D) such that each vertex of D lies in an (u, v)-geodesic, for some u, v ∈ S. The cardinality of a minimum geodetic set is the geodetic number of D and it is denoted by gn→(D).In this work, we first present an upper bound for the hull number of oriented split graphs. Then, we turn our attention to the computational complexity of determining such parameters. We first show that computing hn→(D) is NP-hard for partial cubes, a subclass of bipartite graphs, and that computing gn→(D) is also NP-hard for directed acyclic graphs (DAG). Finally, we present a positive result by showing how to compute such parameters in polynomial time when the input graph is an oriented cactus.

Optimization for Onshore Wind Farm Cable: Connection Layout using ACO-AIA algorithm

The wind farm layout optimization problem is similar to the classical mathematical problem of finding the Steiner Minimal Tree Problem (SMTP) of a weighted undirected graph. Due to the cable current-carrying capacity limitation, the cable sectional area should be carefully selected to meet the system operational requirement and this constraint should be considered during the SMTP formulation process. Hence, traditional SMTP algorithm cannot ensure a minimal cable investment layout. In this paper, a hybrid algorithm based on modified Ants Colony Optimization (ACO) and Artificial Immune Algorithm (AIA) for solving SMTP is introduced. Since the Steiner Tree Problem is NP-hard, we design an algorithm to construct high quality Steiner trees in a short time which is suitable for real time multicast routing in networks. After the breadth - first traversal of the minimal graph obtained by ACO, the terminal points are divided into different convex hull sets, and the full Steiner tree is structured from the convex hull sets partition. The Steiner points can then be vaccinated by AIA to get an optimal graph. The average optimization effect of AIA is shorter than the minimal graph obtained using ACO, and the performance of the algorithm is shown. We give an example of application in optimization for onshore wind farm Cable. The possibility of using different sectional area’s cable is also considered in this paper.

Location Privacy in Mobile Big Data: User Identifiability via Habitat Region Representation

Mobile big data collected by mobile network operators is of interest to many research communities and industries for its remarkable values. However, such spatiotemporal information may lead to a harsh threat to subscribers’ privacy. This work focuses on subscriber privacy vulnerability assessment in terms of user identifiability across two datasets with significant detail reduced mobility representation. In this paper, we propose an innovative semantic spatiotemporal representation for each subscriber based on the geographic information, termed as daily habitat region, to approximate the subscriber’s daily mobility coverage with far lesser information compared with original mobility traces. The daily habitat region is realized via convex hull extraction on the user’s daily spatiotemporal traces. As a result, user identification can be formulated to match two records with the maximum similarity score between two convex hull sets, obtained by our proposed similarity measures based on cosine distance and permutation hypothesis test. Experiments are conducted to evaluate our proposed innovative mobility representation and user identification algorithms, which also demonstrate that the subscriber’s mobile privacy is under a severe threat even with significantly reduced spatiotemporal information.

Two classes of graphs in which some problems related to convexity are efficiently solvable

Monophonic, geodesic and 2-geodesic convexities ([Formula: see text]-convexity, [Formula: see text]-convexity and [Formula: see text]-convexity, for short) on graphs are based on the families of induced paths, shortest paths and shortest paths of length [Formula: see text], respectively. We introduce a class of graphs, the class of cross-cyclicgraphs, in which every connected [Formula: see text]-convex set is also [Formula: see text]-convex and [Formula: see text]-convex. We show that this class is properly contained in the class, say [Formula: see text], of graphs in which geodesic and monophonic convexities are equivalent and properly contains the class of distance-hereditary graphs. Moreover, we show that (1) an [Formula: see text]-hull set (i.e., a subset of vertices, with minimum cardinality, whose [Formula: see text]-convex hull equals the whole vertex set) and, hence, the m-hull number and the g-hull number of a graph in [Formula: see text] can be computed in polynomial time and that (2) both the geodesic-convex hull and the monophonic-convex hull can be computed in linear time in a cross-cyclic graph without cycles of length [Formula: see text] and, hence, in a bipartite distance-hereditary graph.

Polynomial Time Algorithms for Computing a Minimum Hull Set in Distance-Hereditary and Chordal Graphs

We give linear and polynomial time algorithms for computing the hull number of distance-hereditary and chordal graphs, respectively. The complexity of computing the hull number in chordal and distance-hereditary graphs has been open since the introduction of the notion in [M. G. Everett and S. B. Seidman, Discrete Math., 57 (1985), pp. 217--223]. Prior to our result polynomial time algorithms were only known for subclasses of considered graph classes, e.g., split graphs, cographs, interval graphs. Our techniques allow us to give at the same time a linear time algorithm for computing the geodetic number in distance-hereditary graphs. Another consequence of the techniques used is an incremental output-polynomial algorithm to list the set of (inclusion-wise) minimal hull sets in any graphs.

The maximum infection time in the geodesic and monophonic convexities

Recent papers investigated the maximum infection time tP3(G) of the P3-convexity (also called maximum time of 2-neighbor bootstrap percolation) and the maximum infection time tmo(G) of the monophonic convexity. In 2014, it was proved that, for bipartite graphs, deciding whether tP3(G)≥k is polynomial time solvable for k≤4, but is NP-complete for k≥5[23]. In 2015, it was proved that deciding whether tmo(G)≥2 is NP-complete even for bipartite graphs [12]. In this paper, we investigate the maximum infection time tgd(G) of the geodesic convexity. We prove that deciding whether tgd(G)≥k is polynomial time solvable for k=1, but is NP-complete for k≥2 even for bipartite graphs. We also present an O(n3m)-time algorithm to determine tgd(G) and tmo(G) in distance-hereditary graphs. For this, we characterize all minimal hull sets of a general graph in the monophonic convexity. Moreover, we improve the complexity of the fastest known algorithm for finding a minimum hull set of a general graph in the monophonic convexity.

Hull number: [formula omitted]-free graphs and reduction rules

In this paper, we study the (geodesic) hull number of graphs. For any two vertices u,v∈V of a connected undirected graph G=(V,E), the closed interval I[u,v] of u and v is the set of vertices that belong to some shortest (u,v)-path. For any S⊆V, let I[S]=⋃u,v∈SI[u,v]. A subset S⊆V is (geodesically) convex if I[S]=S. Given a subset S⊆V, the convex hull Ih[S] of S is the smallest convex set that contains S. We say that S is a hull set of G if Ih[S]=V. The size of a minimum hull set of G is the hull number of G, denoted by hn(G).First, we show a polynomial-time algorithm to compute the hull number of any P5-free triangle-free graph. Then, we present four reduction rules based on vertices with the same neighborhood. We use these reduction rules to propose a fixed parameter tractable algorithm to compute the hull number of any graph G, where the parameter can be the size of a vertex cover of G or, more generally, its neighborhood diversity, and we also use these reductions to characterize the hull number of the lexicographic product of any two graphs.

Optimization of Cable Layout Design in a Wind Farm: A Hybrid Approach

In this paper, a hybrid algorithm based on modified Ants Colony Optimization (ACO) and Artificial Immune Algorithm (AIA) for solving the Steiner Minimal Tree Problem (SMTP) is introduced. Since the Steiner Tree Problem is NP-hard, we design an algorithm to construct high quality Steiner trees in a short time which is suitable for real time multicast routing in networks. After the breadth - first traversal of the minimal graph obtained by ACO, the terminal points are divided into different convex hull sets, and the full Steiner tree is structured from the convex hull sets partition. The Steiner points can be vaccinated by AIA to get an optimal graph. The average optimization effect of AIA is shorter than the minimal graph obtained using ACO, and the performance of the algorithm is shown. We give an example of application in wind farm network design.

Block decomposition approach to compute a minimum geodetic set

In this paper, we develop a divide-and-conquer approach, called block decomposition, to solve the minimum geodetic set problem. This provides us with a unified approach for all graphs admitting blocks for which the problem of finding a minimum geodetic set containing a given set of vertices (g -extension problem) can be efficiently solved. Our method allows us to derive linear time algorithms for the minimum geodetic set problem in (a proper superclass of) block-cacti and monopolar chordal graphs. Also, we show that hull sets and geodetic sets of block-cacti are the same, and the minimum geodetic set problem is NP-hard in cobipartite graphs. We conclude by pointing out several interesting research directions.

On Minimal and Minimum Hull Sets

A graph convexity (G,C) is a graph G together with a collection C of subsets of V(G), called convex sets, such that ∅,V(G)∈C and C is closed under intersections. For a set U⊆V(G), the hull of U, denoted H(U), is the smallest convex set containing U. If H(U)=V(G), then U is a hull set of G. Motivated by the theory of well covered graphs, which investigates the relation between maximal and maximum independent sets of a graph, we study the relation between minimal and minimum hull sets. We concentrate on the P3 convexity, where convex sets are closed under adding common neighbors of their elements.

Hull number: [formula omitted]-free graphs and reduction rules

In this paper, we study the (geodesic) hull number of graphs. For any two vertices u,v∈V of a connected undirected graph G=(V,E), the closed interval I[u,v] of u and v is the set of vertices that belong to some shortest (u,v)-path. For any S⊆V, let I[S]=⋃u,v∈SI[u,v]. A subset S⊆V is (geodesically) convex if I[S]=S. Given a subset S⊆V the convex hull Ih[S] of S is the smallest convex set that contains S. We say that S is a hull set of G if Ih[S]=V. The size of a minimum hull set of G is the hull number of G, denoted by hn(G).First, we show a polynomial-time algorithm to compute the hull number of any P5-free triangle-free graph. Then, we present four reduction rules based on vertices with the same neighborhood. We use these reduction rules to propose a fixed parameter tractable algorithm to compute the hull number of any graph G, where the parameter can be the size of a vertex cover of G or, more generally, its neighborhood diversity, and we also use these reductions to characterize the hull number of the lexicographic product of any two graphs.

On the hull number of some graph classes

In this paper, we study the geodetic convexity of graphs, focusing on the problem of the complexity of computing a minimum hull set of a graph in several graph classes.For any two vertices u,v∈V of a connected graph G=(V,E), the closed interval I[u,v] of u and v is the set of vertices that belong to some shortest (u,v)-path. For any S⊆V, let I[S]=⋃u,v∈SI[u,v]. A subset S⊆V is geodesically convex or convex if I[S]=S. In other words, a subset S is convex if, for any u,v∈S and for any shortest (u,v)-path P, V(P)⊆S. Given a subset S⊆V, the convex hull Ih[S] of S is the smallest convex set that contains S. We say that S is a hull set of G if Ih[S]=V. The size of a minimum hull set of G is the hull number of G, denoted by hn(G). The Hull Number problem is to decide whether hn(G)≤k, for a given graph G and an integer k. Dourado et al. showed that this problem is NP-complete in general graphs.In this paper, we answer an open question of Dourado et al. (2009) [1] by showing that the Hull Number problem is NP-hard even when restricted to the class of bipartite graphs. Then, we design polynomial-time algorithms to solve the Hull Number problem in several graph classes. First, we deal with the class of complements of bipartite graphs. Then, we generalize some results in Araujo et al. (2011) [2] to the class of (q,q−4)-graphs and to cacti. Finally, we prove tight upper bounds on the hull numbers. In particular, we show that the hull number of an n-node graph G without simplicial vertices is at most 1+⌈3(n−1)5⌉ in general, at most 1+⌈n−12⌉ if G is regular or has no triangle, and at most 1+⌈n−13⌉ if G has girth at least 6.

Gain-Scheduled Worst-Case Control on Nonlinear Stochastic Systems Subject to Actuator Saturation and Unknown Information

In this paper, we propose a method for designing continuous gain-scheduled worst-case controller for a class of stochastic nonlinear systems under actuator saturation and unknown information. The stochastic nonlinear system under study is governed by a finite-state Markov process, but with partially known jump rate from one mode to another. Initially, a gradient linearization procedure is applied to describe such nonlinear systems by several model-based linear systems. Next, by investigating a convex hull set, the actuator saturation is transferred into several linear controllers. Moreover, worst-case controllers are established for each linear model in terms of linear matrix inequalities. Finally, a continuous gain-scheduled approach is employed to design continuous nonlinear controllers for the whole nonlinear jump system. A numerical example is given to illustrate the effectiveness of the developed techniques.