scholarly journals On P_4-tidy graphs

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
V. Giakoumakis ◽  
F. Roussel ◽  
H. Thuillier
Keyword(s):  
Optimization Problems ◽  
Linear Time ◽  
Hamiltonian Path ◽  
Clique Number ◽  
Stability Number ◽  
Sparse Graphs ◽  
Modular Decomposition ◽  
New Class ◽  
International Audience ◽  
Linear Algorithms

International audience We study the P_4-tidy graphs, a new class defined by Rusu [30] in order to illustrate the notion of P_4-domination in perfect graphs. This class strictly contains the P_4-extendible graphs and the P_4-lite graphs defined by Jamison & Olariu in [19] and [23] and we show that the P_4-tidy graphs and P_4-lite graphs are closely related. Note that the class of P_4-lite graphs is a class of brittle graphs strictly containing the P_4-sparse graphs defined by Hoang in [14]. McConnel & Spinrad [2] and independently Cournier & Habib [5] have shown that the modular decomposition tree of any graph is computable in linear time. For recognizing in linear time P_4-tidy graphs, we apply a method introduced by Giakoumakis in [9] and Giakoumakis & Fouquet in [6] using modular decomposition of graphs and we propose linear algorithms for optimization problems on such graphs, as clique number, stability number, chromatic number and scattering number. We show that the Hamiltonian Path Problem is linear for this class of graphs. Our study unifies and generalizes previous results of Jamison & Olariu ([18], [21], [22]), Hochstattler & Schindler[16], Jung [25] and Hochstattler & Tinhofer [15].

ON GRAPHS WITH LIMITED NUMBER OF P4-PARTNERS

1999 ◽  
Vol 10 (01) ◽  
pp. 103-121 ◽  
Author(s):  
FLORIAN ROUSSEL ◽  
IRENA RUSU ◽  
HENRI THUILLIER
Keyword(s):  
Chromatic Number ◽  
Optimization Problems ◽  
Linear Time ◽  
Clique Number ◽  
Recognition Algorithm ◽  
Stability Number

The study of graphs containing few P4's generated an important number of results related to perfection, recognition, optimization problems (see [12], [15], [8]). We define here a new, larger class of graphs and show that the indicated problems may be efficiently solved on this class too (thus generalizing some of the previous results). Namely, we give a linear time recognition algorithm for this class and we note that the optimization problems concerning the clique number, stability number, chromatic number and clique cover number are solvable in linear time.

scholarly journals Spanning connectedness and Hamiltonian thickness of graphs and interval graphs

2015 ◽  
Vol Vol. 16 no. 2 (PRIMA 2013) ◽  
Author(s):  
Peng Li ◽  
Yaokun Wu
Keyword(s):  
Linear Time ◽  
Hamiltonian Path ◽  
Interval Graphs ◽  
Spanning Subgraph ◽  
Linear Forest ◽  
Connectedness Property ◽  
Vertex Ordering ◽  
International Audience ◽  
Vertex Disjoint ◽  
Linear Time Algorithms

International audience A spanning connectedness property is one which involves the robust existence of a spanning subgraph which is of some special form, say a Hamiltonian cycle in which a sequence of vertices appear in an arbitrarily given ordering, or a Hamiltonian path in the subgraph obtained by deleting any three vertices, or three internally-vertex-disjoint paths with any given endpoints such that the three paths meet every vertex of the graph and cover the edges of an almost arbitrarily given linear forest of a certain fixed size. Let π = π1 · · · πn be an ordering of the vertices of an n-vertex graph G. For any positive integer k ≤ n − 1, we call π a k-thick Hamiltonian vertex ordering of G provided it holds for all i ∈ {1,. .. , n − 1} that πiπi+1 ∈ E(G) and the number of neighbors of πi among {πi+1,. .. , πn} is at least min{n − i, k}; For any nonnegative integer k, we say that π is a −k-thick Hamiltonian vertex ordering of G provided |{i : πiπi+1 / ∈ E(G), 1 ≤ i ≤ n − 1}| ≤ k + 1. Our main discovery is that the existence of a thick Hamiltonian vertex ordering will guarantee that the graph has various kinds of spanning connectedness properties and that for interval graphs, quite a few seemingly unrelated spanning connectedness properties are equivalent to the existence of a thick Hamiltonian vertex ordering. Due to the connection between Hamiltonian thickness and spanning connectedness properties, we can present several linear time algorithms for associated problems. This paper suggests that much work in graph theory may have a spanning version which deserves further study, and that the Hamiltonian thickness may be a useful concept in understanding many spanning connectedness properties.

scholarly journals Getting new algorithmic results by extending distance-hereditary graphs via split composition

2021 ◽  
Vol 7 ◽  
pp. e627
Author(s):  
Serafino Cicerone ◽  
Gabriele Di Stefano
Keyword(s):  
Chromatic Number ◽  
Domination Number ◽  
Graph Isomorphism ◽  
Clique Number ◽  
Efficient Algorithms ◽  
Combinatorial Problems ◽  
Stability Number ◽  
New Class ◽  
Graph Classes ◽  
Graph Class

In this paper, we consider the graph class denoted as Gen(∗;P3,C3,C5). It contains all graphs that can be generated by the split composition operation using path P3, cycle C3, and any cycle C5 as components. This graph class extends the well-known class of distance-hereditary graphs, which corresponds, according to the adopted generative notation, to Gen(∗;P3,C3). We also use the concept of stretch number for providing a relationship between Gen(∗;P3,C3) and the hierarchy formed by the graph classes DH(k), with k ≥1, where DH(1) also coincides with the class of distance-hereditary graphs. For the addressed graph class, we prove there exist efficient algorithms for several basic combinatorial problems, like recognition, stretch number, stability number, clique number, domination number, chromatic number, and graph isomorphism. We also prove that graphs in the new class have bounded clique-width.

scholarly journals Duality Theorems for (ρ,ψ,d)-Quasiinvex Multiobjective Optimization Problems with Interval-Valued Components

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 894
Author(s):  
Savin Treanţă
Keyword(s):  
Multiobjective Optimization ◽  
Optimization Problems ◽  
Integral Functional ◽  
Multiple Integral ◽  
Control Problems ◽  
Duality Theorems ◽  
New Class ◽  
Duality Results ◽  
Multiobjective Optimization Problems ◽  
Interval Valued

The present paper deals with a duality study associated with a new class of multiobjective optimization problems that include the interval-valued components of the ratio vector. More precisely, by using the new notion of (ρ,ψ,d)-quasiinvexity associated with an interval-valued multiple-integral functional, we formulate and prove weak, strong, and converse duality results for the considered class of variational control problems.

Diagnosability and Diagnostic Algorithm for Pancake Graph under the Comparison Model

2015 ◽  
Vol 15 (01n02) ◽  
pp. 1550005
Author(s):  
WENJUN LIU ◽  
CHENG-KUAN LIN
Keyword(s):  
Fault Diagnosis ◽  
Interconnection Networks ◽  
Linear Time ◽  
Hamiltonian Path ◽  
Diagnostic Algorithm ◽  
Efficient Algorithms ◽  
Comparison Model ◽  
Diagnosis Model

Fault diagnosis is important for the reliability of interconnection networks. This paper addresses the fault diagnosis of n-dimensional pancake graph Pn under the comparison diagnosis model. By the concept of local diagnosability, we first prove that the diagnosabitly of Pn is n − 1, and it has strong local diagnosability property even if there are n − 3 faulty edges. Furthermore, we present efficient algorithms to locate extended star and Hamiltonian path structures in Pn, respectively. According to the works of Li et al. and Lai, the extended star and Hamiltonian path structures can be used to identify all faulty vertices in linear time, provided the number of faulty vertices is no more than n − 1.

scholarly journals Подход к решению задачи оптимизации логистики агрохолдинга

2013 ◽  
pp. 139-144
Author(s):  
Е. В. Скакалина
Keyword(s):  
Information Technology ◽  
Discrete Optimization ◽  
Optimization Problems ◽  
Implementation Process ◽  
Optimal Solutions ◽  
New Class ◽  
Modern Computer ◽  
Foreign Countries ◽  
Intensive Development ◽  
Discrete Optimization Problems

У роботі наведено короткий аналіз використання інформаційних технологій в аграрному напрямі. Вказу-ється на можливість удосконалення управління проце-сом реалізації логістики великих агрохолдингів за раху-нок використання ефективного методу побудови оп-тимальних рішень для узагальнень задачі про призна-чення. Представлений новий клас дискретних оптимі-заційних задач. Звертається увага на інтенсивний роз-виток логістики у зарубіжних країнах на основі викори-стання сучасних комп'ютерних технологій. The paper presents a brief analysis of the use of information technology in the agricultural area. The possibility of improvement of management of logistics implementation process of large agricultural holdings through the use of an effective method of optimal solutions constructing for generalizations of the assignment problem is shown. A new class of discrete optimization problems is presented. The attention is drawn to the intensive development of logistics in foreign countries on the basis of use of modern computer technologies.

scholarly journals Graphs with many vertex-disjoint cycles

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Dieter Rautenbach ◽  
Friedrich Regen
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Linear Time ◽  
Simple Extension ◽  
Disjoint Cycles ◽  
Cyclomatic Number ◽  
Finite Set ◽  
Extension Rule ◽  
International Audience ◽  
Vertex Disjoint

Graph Theory International audience We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

scholarly journals Entropy-Penalized Semidefinite Programming

2019 ◽  
Author(s):  
Mikhail Krechetov ◽  
Jakub Marecek ◽  
Yury Maximov ◽  
Martin Takac
Keyword(s):  
Machine Learning ◽  
Time Complexity ◽  
Optimization Problems ◽  
Linear Time ◽  
Broad Class ◽  
Low Rank ◽  
Learning Problems ◽  
Unified Framework ◽  
Gradient Computation ◽  
Machine Learning Applications

Low-rank methods for semi-definite programming (SDP) have gained a lot of interest recently, especially in machine learning applications. Their analysis often involves determinant-based or Schatten-norm penalties, which are difficult to implement in practice due to high computational efforts. In this paper, we propose Entropy-Penalized Semi-Definite Programming (EP-SDP), which provides a unified framework for a broad class of penalty functions used in practice to promote a low-rank solution. We show that EP-SDP problems admit an efficient numerical algorithm, having (almost) linear time complexity of the gradient computation; this makes it useful for many machine learning and optimization problems. We illustrate the practical efficiency of our approach on several combinatorial optimization and machine learning problems.

scholarly journals Pebble Game Algorithms and (k,l)-Sparse Graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Audrey Lee ◽  
Ileana Streinu
Keyword(s):  
Efficient Algorithms ◽  
Sparse Graphs ◽  
Edge Disjoint ◽  
International Audience ◽  
Tree Decompositions ◽  
Pebble Game

International audience A multi-graph $G$ on n vertices is $(k,l)$-sparse if every subset of $n'≤n$ vertices spans at most $kn'-l$ edges, $0 ≤l < 2k$. $G$ is tight if, in addition, it has exactly $kn - l$ edges. We characterize $(k,l)$-sparse graphs via a family of simple, elegant and efficient algorithms called the $(k,l)$-pebble games. As applications, we use the pebble games for computing components (maximal tight subgraphs) in sparse graphs, to obtain inductive (Henneberg) constructions, and, when $l=k$, edge-disjoint tree decompositions.

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