Series-parallel Graphs Research Articles

Circular Chromatic Number of Signed Graphs

A signed graph is a pair $(G, \sigma)$, where $G$ is a graph (loops and multi edges allowed) and $\sigma: E(G) \to \{+, -\}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend circular coloring of graphs to signed graphs. Given a signed graph $(G, \sigma)$ with no positive loop, a circular $r$-coloring of $(G, \sigma)$ is an assignment $\psi$ of points of a circle of circumference $r$ to the vertices of $G$ such that for every edge $e=uv$ of $G$, if $\sigma(e)=+$, then $\psi(u)$ and $\psi(v)$ have distance at least $1$, and if $\sigma(e)=-$, then $\psi(v)$ and the antipodal of $\psi(u)$ have distance at least $1$. The circular chromatic number $\chi_c(G, \sigma)$ of a signed graph $(G, \sigma)$ is the infimum of those $r$ for which $(G, \sigma)$ admits a circular $r$-coloring. For a graph $G$, we define the signed circular chromatic number of $G$ to be $\max\{\chi_c(G, \sigma): \sigma \text{ is a signature of $G$}\}$. 
 We study basic properties of circular coloring of signed graphs and develop tools for calculating $\chi_c(G, \sigma)$. We explore the relation between the circular chromatic number and the signed circular chromatic number of graphs, and present bounds for the signed circular chromatic number of some families of graphs. In particular, we determine the supremum of the signed circular chromatic number of $k$-chromatic graphs of large girth, of simple bipartite planar graphs, $d$-degenerate graphs, simple outerplanar graphs and series-parallel graphs. We construct a signed planar simple graph whose circular chromatic number is $4+\frac{2}{3}$. This is based and improves on a signed graph built by Kardos and Narboni as a counterexample to a conjecture of Máčajová, Raspaud, and Škoviera.

Confluence up to garbage in graph transformation

The transformation of graphs and graph-like structures is ubiquitous in computer science. When a system is described by graph-transformation rules, it is often desirable that the rules are both terminating and confluent so that rule applications in an arbitrary order produce unique resulting graphs. However, there are application scenarios where the rules are not globally confluent but confluent on a subclass of graphs that are of interest. In other words, non-resolvable conflicts can only occur on graphs that are considered as “garbage”. In this paper, we introduce the notion of confluence up to garbage and generalise Plump's critical pair lemma for double-pushout graph transformation, providing a sufficient condition for confluence up to garbage by non-garbage critical pair analysis. We apply our results in two case studies about efficient language recognition: we present backtracking-free graph reduction systems which recognise a class of flow diagrams and a class of labelled series-parallel graphs, respectively. Both systems are non-confluent but confluent up to garbage. We also give a critical pair condition for subcommutativity up to garbage which, together with closedness, implies confluence up to garbage even in non-terminating systems.

A note on integral generalized flows in directed partial 2-trees

In 2013 Krumke and Zeck developed a pseudopolynomial algorithm to compute maximal integral generalized flows in 2-terminal series-parallel digraphs. In this note we observe that this approach can be extended to the more general class of k-terminal directed partial 2-trees.

Passive mechanical realizations of bicubic impedances with no more than five elements for inerter-based control design

This paper mainly investigates the passive realization problems of bicubic (third-order) impedances as damper-spring-inerter networks consisting of no more than five elements. First, the special case where a bicubic impedance contains a pole or a zero on the imaginary axis or at infinity is discussed. Then, assuming that there is no pole or zero on the imaginary axis or at infinity, the realizations of bicubic impedances as five-element networks are investigated. Necessary and sufficient conditions for the realizability as five-element series-parallel networks and as five-element non-series-parallel networks are derived, respectively, where 22 series-parallel configurations and 11 non-series-parallel configurations are presented to cover the conditions. Finally, two numerical examples together with positive-real controller designs for a quarter-car suspension system are presented for illustrations. The results of this paper can contribute to the synthesis of low-complexity passive mechanical (or electrical) networks, which are motivated by the synthesis and design of inerter-based vibration control systems.

Refinement of the Asymptotics of the Number of Labeled Series-Parallel Graphs

Refinement of the Asymptotics of the Number of Labeled Series-Parallel Graphs

Concurrence Percolation in Quantum Networks.

Establishing long-distance quantum entanglement, i.e., entanglement transmission, in quantum networks (QN) is a key and timely challenge for developing efficient quantum communication. Traditional comprehension based on classical percolation assumes a necessary condition for successful entanglement transmission between any two infinitely distant nodes: they must be connected by at least a path of perfectly entangled states (singlets). Here, we relax this condition by explicitly showing that one can focus not on optimally converting singlets but on establishing concurrence-a key measure of bipartite entanglement. We thereby introduce a new statistical theory, concurrence percolation theory (ConPT), remotely analogous to classical percolation but fundamentally different, built by generalizing bond percolation in terms of "sponge-crossing" paths instead of clusters. Inspired by resistance network analysis, we determine the path connectivity by series and parallel rules and approximate higher-order rules via star-mesh transforms. Interestingly, we find that the entanglement transmission threshold predicted by ConPT is lower than the known classical-percolation-based results and is readily achievable on any series-parallel networks such as the Bethe lattice. ConPT promotes our understanding of how well quantum communication can be further systematically improved versus classical statistical predictions under the limitation of QN locality-a "quantum advantage" that is more general and efficient than expected. ConPT also shows a percolationlike universal critical behavior derived by finite-size analysis on the Bethe lattice and regular two-dimensional lattices, offering new perspectives for a theory of criticality in entanglement statistics.

The conjunction of the linear arboricity conjecture and Lovász's path partition theorem

A graph is a linear forest if each of its components is a path. Given a graph G with maximum degree Δ(G), motivated by the famous linear arboricity conjecture and Lovász's classic result on partitioning the edge set of a graph into paths, we call a partition F:=F1|⋯|Fk of the edge set of G an exact linear forest partition if each Fi induces a linear forest, k≤⌈Δ(G)+12⌉, and every vertex v∈V(G) is on at most ⌈dG(v)+12⌉ non-trivial paths belonging to F. In this paper, we prove the following two results.•Every 2-degenerate graph has an exact linear forest partition, and so does every series-parallel graph, every outerplanar graph, and every subdivision of any graph provided each edge of the original graph is subdivided at least once.•Let p∈(0,1) be a constant. If G∼Gn,p, then a.a.s. G has an exact linear forest partition.

Partial duality for ribbon graphs, II: Partial-twuality polynomials and monodromy computations

The partial (Poincaré) dual with respect to a subset A of edges of a ribbon graph G was introduced by Chmutov in connection with the Jones–Kauffman and Bollobás–Riordan polynomials. In developing the theory of maps, Wilson and others have composed Poincaré duality ∗ with Petrie duality × to give Wilson duality ∗×∗ and two trialities∗× and ×∗. In further expanding the theory, Abrams and Ellis-Monaghan have called the five operators twualities. Part I of this investigation (Gross et al., 2020) introduced as a partial-∗polynomial of G, the generating function enumerating partial Poincaré duals by Euler-genus. In this sequel, we introduce the corresponding partial-×, -∗×, -×∗, and -∗×∗polynomials for their respective twualities. For purposes of computation, we express each partial twuality in terms of the monodromy of permutations of the flags of a map. We analyze how single-edge partial twualities affect the three types (proper, untwisted, twisted) of edges. Various possible properties of partial-twuality polynomials are studied, including interpolation and log-concavity; machine-computed unimodal counterexamples to some log-concavity conjectures from Gross et al. (2020) are given. It is shown that the partial-∗×∗ polynomial for a ribbon graph G equals the partial-× polynomial for G∗. Formulas or recursions are given for various families of graphs, including ladders and, for Wilson duality, a large subfamily of series–parallel networks. All of these polynomials are shown to be log-concave.

A polynomial time algorithm to compute the connected treewidth of a series–parallel graph

It is well known that the treewidth of a graph G corresponds to the node search number where a team of searchers is pursuing a fugitive that is lazy and invisible (or alternatively is agile and visible) and has the ability to move with infinite speed via unguarded paths. Recently, monotone and connected node search strategies have been considered. A search strategy is monotone if it prevents the fugitive from pervading again areas from where he had been expelled and is connected if, at each step, the set of vertices that is or has been occupied by the searchers induces a connected subgraph of G. It has been shown that the corresponding connected and monotone search number of a graph G can be expressed as the connected treewidth, denoted by ctw(G), that is defined as the minimum width of a rooted tree-decomposition (X,T,r), where the union of the bags corresponding to the nodes of a path of T containing the root r is connected in G. In this paper, we initiate the algorithmic study of connected treewidth. We design a O(n2⋅logn)-time dynamic programming algorithm to compute the connected treewidth of biconnected series–parallel graphs. At the price of an extra n factor in the running time, our algorithm generalizes to graphs of treewidth at most two.

The Minimum Tollbooth Problem in Atomic Network Congestion Games with Unsplittable Flows

This work analyzes the minimum tollbooth problem in atomic network congestion games with unsplittable flows. The goal is to place tolls on edges, such that there exists a pure Nash equilibrium in the tolled game that is a social optimum in the untolled one. Additionally, we require the number of tolled edges to be the minimum. This problem has been extensively studied in non-atomic games, however, to the best of our knowledge, it has not been considered for atomic games before. By a reduction from the weighted CNF SAT problem, we show both the NP-hardness of the problem and the W[2]-hardness when parameterizing the problem with the number of tolled edges. On the positive side, we present a polynomial time algorithm for networks on series-parallel graphs that turns any given state of the untolled game into a pure Nash equilibrium of the tolled game with the minimum number of tolled edges.

Typical Sequences Revisited — Computing Width Parameters of Graphs

In this work, we give a structural lemma on merges of typical sequences, a notion that was introduced in 1991 [Lagergren and Arnborg, Bodlaender and Kloks, both ICALP 1991] to obtain constructive linear time parameterized algorithms for treewidth and pathwidth. The lemma addresses a runtime bottleneck in those algorithms but so far it does not lead to asymptotically faster algorithms. However, we apply the lemma to show that the cutwidth and the modified cutwidth of series parallel digraphs can be computed in polynomial time.

On the Enumeration of Labeled Series-Parallel $$k $$-Cyclic $$2 $$-Connected Graphs

We deduce an explicit formula for the number of labeled series-parallel $$k $$ -cyclic $$n $$ -vertex $$2 $$ -connected graphs and find the corresponding asymptotics for a large number of vertices and a fixed $$k $$ . Under the uniform probability distribution, an asymptotic formula is obtained for the probability that a random $$n $$ -vertex $$k $$ -cyclic $$2 $$ -connected graph with a large number of vertices and a fixed $$k $$ is a series-parallel graph.

Treelength of Series-parallel Graphs

The length of a tree-decomposition of a graph is the maximum distance between two vertices of a same bag of the decomposition. The treelength of a graph is the minimum length among its tree-decompositions. Treelength of graphs has been studied for its algorithmic applications in classical metric problems such as Traveling Salesman Problem or metric dimension of graphs and also, in compact routing in the context of distributed computing. Deciding whether the treelength of a general graph is at most 2 is NP-complete (graphs of treelength one are precisely the chordal graphs), and it is known that the treelength of a graph cannot be approximated up to a factor less than 3/2 (the best known approximation algorithm for treelength has an approximation ratio of 3). However, nothing is known on the computational complexity of treelength in planar graphs, except that the treelength of any outerplanar graph is equal to the third of the maximum size of its isometric cycles. This work initiates the study of treelength in planar graphs by considering the next natural superclass of outerplanar graphs, namely the one of series-parallel graphs.We first fully describe the treelength of melon graphs (set of pairwise internally disjoint paths linking two vertices), showing that, even in such a restricted graph class, the expression of the treelength is not trivial. Then, we show that treelength can be approximated up to a factor 3/2 in series-parallel graphs. Our main result is a polynomial-time algorithm for deciding whether a series-parallel graph has treelength at most 2. Our latter result relies on a characterization of series-parallel graphs with treelength 2 in terms of infinite families of forbidden isometric subgraphs.

Characterizing 2-Trees Relative to Chordal and Series-Parallel Graphs

The 2-connected 2-tree graphs are defined as being constructible from a single 3-cycle by recursively appending new degree-2 vertices so as to form 3-cycles that have unique edges in common with the existing graph. Such 2-trees can be characterized both as the edge-minimal chordal graphs and also as the edge-maximal series-parallel graphs. These are also precisely the 2-connected graphs that are simultaneously chordal and series-parallel, where these latter two better-known types of graphs have themselves been both characterized and applied in numerous ways that are unmotivated by their interaction with 2-trees and with each other.

Dynamic DAG Scheduling Under Memory Constraints for Shared-Memory Platforms

This work focuses on dynamic DAG scheduling under memory constraints. We target a shared-memory platform equipped with $p$ parallel processors. The goal is to bound the maximum amount of memory that may be needed by any schedule using p processors to execute the DAG. We refine the classical model that computes maximum cuts by introducing two types of memory edges in the DAG, black edges for regular precedence constraints and red edges for actual memory consumption during execution. A valid edge cut cannot include more than $p$ red edges. This limitation had never been taken into account in previous works, and dramatically changes the complexity of the problem, which was polynomial and becomes NP-hard. We introduce an Integer Linear Program (ILP) to solve it, together with an efficient heuristic based on rounding the rational solution of the ILP. In addition, we propose an exact polynomial algorithm for series-parallel graphs. We further study the extension of the approach where the scheduler is dynamically constrained to select tasks (among ready tasks) so that the total memory used does not exceed some threshold. We provide an extensive set of experiments, both with randomly-generated graphs and with graphs arising from practical applications, which demonstrate the impact of resource constraints on peak memory usage.

Bipartite and Series-Parallel Graphs Without Planar Lombardi Drawings

We find a family of planar bipartite graphs all of whose Lombardi drawings (drawings with circular arcs for edges, meeting at equal angles at the vertices) are nonplanar. We also find families of embedded series-parallel graphs and apex-trees (graphs formed by adding one vertex to a tree) for which there is no planar Lombardi drawing consistent with the given embedding.

${\mathcal {H}_2}$ Performance of Series-Parallel Networks: A Compositional Perspective

We examine the $\mathcal {H}_2$ norm of matrix-weighted leader-follower consensus on series-parallel networks. By using an extension of electrical network theory on matrix-valued resistances, voltages, and currents, we show that the computation of the ${\mathcal {H}_2}$ norm can be performed efficiently by decomposing the network into atomic elements and composition rules. Finally, we examine the problem of efficiently adapting the matrix-valued edge weights to optimize the ${\mathcal {H}_2}$ norm of the network.

Minimal Series–Parallel Network Realizations of Bicubic Impedances

An important open problem in the synthesis of passive controllers is to obtain a passive network that realizes an arbitrary given impedance function and contains the least possible number of elements. This problem has its origins in electric circuit theory, and is directly applicable to the cost-effective design of mechanical systems containing the inerter. Despite a rich history, the problem can only be considered solved for networks that contain at most two energy storage elements, and in a small number of other special cases. In this article, we solve the minimal network realization problem for the class of impedances realized by series–parallel networks containing at most three energy storage elements. To accomplish this, we develop a novel continuity-based approach to eliminate redundant elements from a network.

Network Flow with Intermediate Storage: Models and Algorithms

Various network flow models, such as a flow maximization, a time minimization, a cost minimization, or a combination of them, have already been investigated. In most of the cases, they are considered subject to the flow conservation constraints. Here, we investigate the network flow models with intermediate storage, i.e., the inflow may be greater than the outflow at intermediate nodes. We introduce a maximum static and a maximum dynamic flow problem where an intermediate storage is allowed. Then, polynomial time algorithms are presented to solve these problems in two terminal general networks. We also study the earliest arrival property of the maximum dynamic flow in two terminal series-parallel networks and present its efficient solution procedure with intermediate storage. Moreover, we introduce a dynamic contraflow model with intermediate storage and present a polynomial time algorithm to solve the maximum dynamic contraflow problem in two terminal networks. We also solve an earliest arrival contraflow problem with intermediate storage. Our investigation is focused to solve the evacuation planning problem where the intermediate storage is permitted.

Non-Greedy Online Steiner Trees on Outerplanar Graphs

This paper addresses the classic online Steiner tree problem on edge-weighted graphs. It is known that a greedy (nearest neighbor) online algorithm has a tight competitive ratio for wide classes of graphs, such as trees, rings, any class including series-parallel graphs, and unweighted graphs with bounded diameter. However, we do not know any greedy or non-greedy tight deterministic algorithm for other classes of graphs. In this paper, we observe that a greedy algorithm is $$\Omega (\log n)$$ -competitive on outerplanar graphs, where n is the number of vertices, and propose a 5.828-competitive deterministic algorithm on outerplanar graphs. Our algorithm connects a requested vertex and the tree constructed thus far using a path that is constant times longer than the distance between them. We also present a lower bound of 4 for arbitrary deterministic online Steiner tree algorithms on outerplanar graphs.