Two-terminal Series-parallel Research Articles

Chronological rectangle digraphs which are two-terminal series–parallel

Chronological interval digraphs are a natural directed analogue of interval graphs. They admit elegant characterizations as well as efficient recognition algorithms. Recently, a generalization of chronological interval digraphs to 2-dimensional space called chronological rectangle digraphs was introduced and studied. Although many interesting properties of chronological rectangle digraphs have been discovered, the most fundamental question of whether this class of digraphs can be recognized in polynomial-time remains unanswered. This is largely due to the lack of a good structural characterization for the class. In this paper, we study chronological rectangle digraphs which are also two-terminal series–parallel. We show that this restricted class of digraphs admits a forbidden induced subdigraph characterization, which leads to a polynomial-time recognition algorithm for this class of digraphs.

Параметры последовательно-параллельных двухполюсников из заданного числа одинаковых элементов

Параметры последовательно-параллельных двухполюсников из заданного числа одинаковых элементов

Series–parallel chromatic hypergraphs

In this paper two-terminal series–parallel chromatic hypergraphs are introduced and for this class of hypergraphs it is shown that the chromatic polynomial can be computed with polynomial complexity. It is also proved that h -uniform multibridge hypergraphs θ ( h ; a 1 , a 2 , … , a k ) are chromatically unique for h ≥ 3 if and only if h = 3 and a 1 = a 2 = ⋯ = a k = 1 , i.e., when they are sunflower hypergraphs having a core of cardinality 2 and all petals being singletons.

Aggregation approach for the minimum binary cost tension problem

The aggregation technique, dedicated to two-terminal series–parallel graphs (TTSP-graphs) and introduced lately to solve the minimum piecewise linear cost tension problem, is adapted here to solve the minimum binary cost tension problem (BCT problem). Even on TTSP-graphs, the BCT problem has been proved to be NP-complete. As far as we know, the aggregation is the only algorithm, with mixed integer programming (MIP), proposed to solve exactly the BCT problem on TTSP-graphs. A comparison of the efficiency of both methods and a heuristic is presented.

Polynomial Time Inductive Inference of TTSP Graph Languages from Positive Data

Two-Terminal Series Parallel (TTSP, for short) graphs are used as data models in applications for electric networks and scheduling problems. We propose a TTSP term graph which is a TTSP graph having structured variables, that is, a graph pattern over a TTSP graph. Let${\\cal TG_{TTSP}}$ be the set of all TTSP term graphs whose variable labels are mutually distinct. For a TTSP term graph g in ${\\cal TG_{TTSP}}$, the TTSP graph language of g, denoted by L(g), is the set of all TTSP graphs obtained from g by substituting arbitrary TTSP graphs for all variables in g. Firstly, when a TTSP graph G and a TTSP term graph g are given as inputs, we present a polynomial time matching algorithm which decides whether or not L(g) contains G. The minimal language problem for the class ${\\cal L_{TTSP}}=\\{L(g) \\mid g\\in {\\cal TG_{TTSP}}\\}$ is, given a set S of TTSP graphs, to find a TTSP term graph g in ${\\cal TG_{TTSP}}$ such that L(g) is minimal among all TTSP graph languages which contain all TTSP graphs in S. Secondly, we give a polynomial time algorithm for solving the minimal language problem for ${\\cal L_{TTSP}}$. Finally, we show that ${\\cal L_{TTSP}}$ is polynomial time inductively inferable from positive data.

Book Embeddability of Series–Parallel Digraphs

In this paper we deal with the problem of computing upward two-page book embeddings of Two Terminal Series-Parallel (TTSP) digraphs, which are a subclass of series-parallel digraphs. An optimal O(n) time and space algorithm to compute an upward two-page book embedding of a TTSP-digraph with n vertices is presented. A previous algorithm of Alzohairi and Rival [1] runs in O(n3) time and assumes that the input series-parallel digraph does not have transitive edges. An application of this result to a computational geometry problem is also discussed. More precisely, upward two-page book embeddings are used to deal with the upward point-set embeddability problem, i.e., the problem of mapping planar digraphs onto a given set of points in the plane so that all edges are monotonically increasing in a common direction. The equivalence between upward two-page book embeddability and upward point-set embeddability with at most one bend per edge on any given set of points is proved. An O(n log n)-time algorithm for computing an upward point-set embedding with at most one bend per edge for TTSP-digraphs is presented.

On the velocity analysis of interconnected chains mechanisms

For parallel mechanisms, a system of linear input–output velocity equations can be obtained after the passive joint velocities are eliminated from the velocity equations via a screw-theory method. This approach can be considered standard. A general method for the elimination of the passive joint velocities, for mechanisms that are not purely parallel, has not yet been proposed. The paper addresses the problem by studying the instantaneous kinematics of two closed-loop mechanisms with kinematic chains not in the two-terminal series-parallel class.

Minimum Dominating Trail Set for Two-Terminal Series Parallel Graphs

Abstract Given a graph G , the Minimum Dominating Trail Set (MDTS) problem consists in finding a minimum cardinality collection of pairwise edge-disjoint trails such that each edge of G has at least one endvertex on some trail. The MDTS problem is NP –hard for general graphs. In this paper an algorithmic approach for the MDTS problem on (two-terminal) series parallel graphs is presented.

An efficient scheme to solve two problems for two-terminal series parallel graphs

An efficient scheme to solve two problems for two-terminal series parallel graphs

A note on the tour problems in two-terminal series-parallel graphs

In this paper, the tour problems in two-terminal series-parallel (TTSP) graphs is studied. A tour in a TTSP graph is a closed walk that passes a set of vertices. Those that pass every vertex at least once are called complete tour. The length of a tour is the summation of lengths of the edges in the tour. The shortest complete tour of a TTSP graph is a complete tour with minimum length. An O(¦E¦) time algorithm to find a shortest complete tour of a TTSP graph is presented, where ¦ E¦ is the number of edges in the graph. It is also shown that the bottleneck two-tour problem in a TTSP graph is NP-complete. The bottleneck two-tour problem in a TTSP graph is to find two tours such that each vertex of this TTSP graph must appear in exactly one of these two tours and the length of the longer tour of these two tours is minimized.

A Semiring on Convex Polygons and Zero-Sum Cycle Problems

Two natural operations on the set of convex polygons are shown to form a closed semiring; the two operations are vector summation and convex hull of the union. Various properties of these operations are investigated. Kleene’s algorithm applied to this closed semiring solves the problem of determining whether a directed graph with two-dimensional labels has a zero-sum cycle or not. This algorithm is shown to run in polynomial time in the special cases of graphs with one-dimensional labels, BTTSP (Backedged Two-Terminal Series-Parallel) graphs, and graphs with bounded labels. The undirected zero-sum cycle problem and the zero-sum simple cycle problem are also investigated.

Binary tree algebraic computation and parallel algorithms for simple graphs

In this paper we define the binary tree algebraic computation (BTAC) problem and develop an efficient parallel algorithm for solving this problem. A variety of graph problems (minimum covering set, minimum r-dominating set, maximum matching set, etc.) for trees and two terminal series parallel (TTSP) graphs can be converted to instances of the BTAC problem. Thus efficient parallel algorithms for these problems are obtained systematically by using the BTAC algorithm. The parallel computation model is an exclusive read exclusive write PRAM. The algorithms for tree problems run in O(log n) time with O( n) processors. The algorithms for TTSP graph problems run in O(log m) time with O( m) processors where n ( m) is the number of vertices (edges) in the input graph. These algorithms are within an O(log n) factor of optimal.

Parallel recognition and decomposition of two terminal series parallel graphs

In this paper, we develop a parallel recognition and decomposition algorithm for two-terminal series parallel (TTSP) graphs. Given a directed acyclic graph G in edge list form, the algorithm determines whether G is a TTSP graph. If G is a TTSP graph, the algorithm constructs a decomposition tree for G . Some interesting properties of the TTSp graphs are derived in order to facilitate fast parallel processing. The algorithm runs in O (log 2 n + log m ) time with O ( n + m ) processors on an exclusive read exclusive write PRAM where n ( m ) is the number of vertices (edges) in G . This algorithm is within a polylogrithmic factor of optimal.

Two-terminal series-parallel networks

This paper discusses two-terminal series-parallel networks occurring in Applied Probability and other contexts where the mathematical theory of reliability is developed. Eighty years ago Mac Mahon successfully discussed the number of different structures built of n identical components [1]; later Knödel [3] and Carlitz and Riordan [4] in the 1950's investigated the number of different structures built of n different components. However, in many new applications, engineers and statisticians have to study structures having n components with a specification defining various types of components and stating how many identical components of each kind should be used. This general problem is investigated in Section 7 whilst in Sections 1–6 results obtained earlier are presented and it is shown there how modern approaches to combinatorics (particularly Pólya's Enumerative Combinatorial Analysis) can simplify some reasoning of previous authors.

Two-terminal series-parallel networks

This paper discusses two-terminal series-parallel networks occurring in Applied Probability and other contexts where the mathematical theory of reliability is developed. Eighty years ago Mac Mahon successfully discussed the number of different structures built of n identical components [1]; later Knödel [3] and Carlitz and Riordan [4] in the 1950's investigated the number of different structures built of n different components. However, in many new applications, engineers and statisticians have to study structures having n components with a specification defining various types of components and stating how many identical components of each kind should be used. This general problem is investigated in Section 7 whilst in Sections 1–6 results obtained earlier are presented and it is shown there how modern approaches to combinatorics (particularly Pólya's Enumerative Combinatorial Analysis) can simplify some reasoning of previous authors.

On The Line‐Group of Two‐Terminal Series‐Parallel Networks

On The Line‐Group of Two‐Terminal Series‐Parallel Networks

The number of labeled two-terminal series-parallel networks

The number of labeled two-terminal series-parallel networks

The Number of Two‐Terminal Series‐Parallel Networks

The Number of Two‐Terminal Series‐Parallel Networks