Series-parallel Graphs Research Articles

Bounding the beta invariant of 3-connected graphs

The beta invariant is closely related to the Chromatic and Tutte Polynomials and has been extensively studied, see Brylawski [A combinatorial model for series-parallel networks, Trans. Amer. Math. Soc. 154 (1971) 1–22], Crapo [A higher invariant for matroids, J. Combin. Theory 2 (1967) 406–417], Lee and Wu [Bounding the beta invariant of 3-connected matroids, Discrete Math. 354 (2022) 1–11], Oxley [On Crapo’s beta invariant for matroids, Stud. Appl. Math. 66(3) (1982) 267–277], and others. Lee and Wu [Bounding the beta invariant of 3-connected matroids, Discrete Math. 354 (2022) 1–11] established that a [Formula: see text]-connected graph with [Formula: see text] vertices possesses a beta invariant of at least [Formula: see text], reaching equality only when the graph is a wheel or the Prism. Additionally, Oxley [On Crapo’s beta invariant for matroids, Stud. Appl. Math. 66(3) (1982) 267–277] provided characterizations for matroids with beta invariant values of two, three, and four. This paper extends the findings of Lee and Wu by offering a comprehensive characterization of all [Formula: see text]-connected graphs with [Formula: see text] vertices that have a beta invariant of either [Formula: see text] or [Formula: see text]. As a consequence, we prove that any [Formula: see text]-connected graph with at least [Formula: see text] vertices other than a Wheel has a beta invariant of at least [Formula: see text].

ON THE TOTAL VERTEX IRREGULARITY STRENGTH OF SERIES PARALLEL GRAPH sp(m,r,4)

his study aims to determine the total vertex irregularity strength on a series parallel graph for and . Total labeling is said to be vertex irregular, if the weights for each vertices are different. Determination of the total vertex irregularity of series parallel graph is done by obtaining the largest lower bound and the smallest upper bound. The lower bound is obtained by analyzing the structure of the graph to obtain the largest minimum label of k and the upper bound is analyzed by labeling the vertices and edges of the graph, where the largest label is k and the values for each vertices weight is different. The result obtained for the total vertex irregularity strength of a series parallel graph is .

Extension Theorems for General Lotto Games With Applications to Network Security

The General Lotto game is a well-studied model where two opposing players strategically allocate a limited amount resources to multiple contests. In the classic setup, each contest represents an individual battlefield with an associated value, and the objective is to maximize the accumulated value by winning individual battlefields. In this paper, we consider scenarios beyond the classic setup, where (i) the notion of a battlefield is generalized to individual contests, (ii) success on a contest can depend on securing subsets of battlefields, and (iii) the winner of a battlefield can be based on alternate winning rules other than the classic winner-take-all rule. Our main results demonstrate that having an equilibrium solution to a single contest scenario can provide immediate equilibrium characterizations for two important generalizations: 1) the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">linear-count extension</i> , where performance is measured by the cumulative weight of the contests that are won. 2) The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">weakest-link extension</i> (best-shot), where one player must win <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">all</i> (at least one) of the constituent contests in order to earn any value. We demonstrate the applicability of these extensions to a general class of network path defense problems. In particular, our results can generate exact solutions to series-parallel networks, as well as provide bounds on equilibrium payoffs bounds for arbitrary networks. To conclude, we provide equilibrium solutions to a novel class of contests with non-linear favoritism winning rules.

A New Formula for the Number of Labeled Series-Parallel Graphs

A New Formula for the Number of Labeled Series-Parallel Graphs

Multi-resource scheduling of moldable workflows

Resource scheduling plays a vital role in High-Performance Computing (HPC) systems. Most scheduling research in HPC has focused on only a single type of resource (e.g., computing cores or I/O resource). With the advancement in hardware architectures and the increase in data-intensive HPC applications, there is a need to simultaneously consider a diverse set of resources (e.g., computing cores, cache, memory, I/O, and network resources) in the design of runtime schedulers for improving the overall application performance. In this paper, we study multi-resource scheduling to minimize the makespan of computational workflows comprised of moldable parallel jobs. Moldable jobs allow the scheduler to flexibly select a variable set of resources before execution, thus can adapt to the available system resources (as compared to rigid jobs) while staying easy to design and implement (as compared to malleable jobs). We propose a Multi-Resource Scheduling Algorithm (MRSA), which combines a novel resource allocation strategy and an extended list scheduling scheme to schedule the jobs. We prove that, on a system with d types of schedulable resources, MRSA achieves an approximation ratio of 1.619d+2.545d+1 for any d≥1, and a ratio of d+3d23+O(d3) when d is large (i.e., d≥22). We also present improved approximation results for workflows comprised of jobs with special precedence constraints (e.g., series-parallel graphs, trees, and independent jobs). Further, we prove a lower bound of d on the approximation ratio of any list-based scheduling algorithm with local priority considerations. Finally, we conduct a comprehensive set of simulations to evaluate the performance of the algorithm using synthetic workflows of different structures and moldable jobs following different speedup models. The results show that MRSA fares better than the theoretical bound predicts, and that it consistently outperforms two baseline heuristics under a variety of parameter settings, illustrating its robust practical performance.

How far is my network from being edge-based? Proximity measures for edge-basedness of unrooted phylogenetic networks

Phylogenetic networks which are, as opposed to trees, suitable to describe processes like hybridization and horizontal gene transfer, play a substantial role in evolutionary research. However, while non-treelike events need to be taken into account, they are relatively rare, which implies that biologically relevant networks are often assumed to be similar to trees in the sense that they can be obtained by taking a tree and adding some additional edges. This observation led to the concept of so-called tree-based networks, which recently gained substantial interest in the literature. Unfortunately, though, identifying such networks in the unrooted case is an NP-complete problem. Therefore, classes of networks for which tree-basedness can be guaranteed are of the utmost interest. The most prominent such class is formed by so-called edge-based networks, which have a close relationship to generalized series–parallel graphs known from graph theory. They can be identified in linear time and are in some regards biologically more plausible than general tree-based networks. While concerning the latter proximity measures for general networks have already been introduced, such measures are not yet available for edge-basedness. This means that for an arbitrary unrooted network, the “distance” to the nearest edge-based network could so far not be determined. The present manuscript fills this gap by introducing two classes of proximity measures for edge-basedness, one based on the given network itself and one based on its so-called leaf shrink graph (LS graph). Both classes contain four different proximity measures, whose similarities and differences we study subsequently.

Burling graphs revisited, part III: Applications to [formula omitted]-boundedness

The Burling sequence is a sequence of triangle-free graphs of unbounded chromatic number. The class of Burling graphs consists of all the induced subgraphs of the graphs of this sequence.In the first and second parts of this work, we introduced derived graphs, a class of graphs, equal to the class of Burling graphs, and proved several geometric and structural results about them.In this third part, we use those results to find some Burling and non-Burling graphs, and we see some applications of this in the theory of χ-boundedness. Specifically, we show that several graphs, like K5, some series-parallel graphs that we call necklaces, and some other graphs are not weakly pervasive.

The circular chromatic number of signed series–parallel graphs of given girth

A signed graph is a graph G together with a signature σ:E(G)→{1,−1}. For a real number r≥1, Cr is a circle of circumference r. For two points a,b∈Cr, the distance d(modr)(a,b) between a and b is the length of the shorter arc of Cr connecting a and b. For x∈Cr, the antipodal x̄ of x is the unique point in Cr of distance r/2 from x. A circular r-colouring of (G,σ) is a mapping f:V(G)→Cr such that for each positive edge e=uv, d(modr)(f(u),f(v))≥1, and for each negative edge e=uv, d(modr)(f(u),f(v)¯)≥1. The circular chromatic number of a signed graph (G,σ) is the minimum r such that (G,σ) is circular r–colourable. Let g∗(G,σ) be the length of the shortest cycle in (G,σ) with an odd number of positive edges. For a positive integer g, let SPg be the family of signed series–parallel graphs with g∗(G,σ)≥g. This paper determines, for any positive integer g, the supremum value of χc(G,σ) of signed graphs in SPg.

Treelength of series–parallel graphs

The length of a tree-decomposition of a graph is the maximum distance (in the graph) 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 32 (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 size of a largest isometric cycle. This work initiates the study of treelength in planar graphs by considering the next natural subclass of planar 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 32 in series–parallel graphs. Our main result is a quadratic-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 an infinite family of forbidden isometric subgraphs.

On the Location of Chromatic Zeros of Series-Parallel Graphs

In this paper we consider the zeros of the chromatic polynomial of series-parallel graphs. Complementing a result of Sokal, giving density outside the disk $|q-1|\leq1$, we show density of these zeros in the half plane $\Re(q)&gt;3/2$ and we show there exists an open region $U$ containing the interval $(0,32/27)$ such that $U\setminus\{1\}$ does not contain zeros of the chromatic polynomial of series-parallel graphs.&#x0D; We also disprove a conjecture of Sokal by showing that for each large enough integer $\Delta$ there exists a series-parallel graph for which all vertices but one have degree at most $\Delta$ and whose chromatic polynomial has a zero with real part exceeding $\Delta$.

Recognizing Series-Parallel Matrices in Linear Time

A series-parallel matrix is a binary matrix that can be obtained from an empty matrix by successively adjoining rows or columns that are copies of an existing row/column or have at most one one-entry. Equivalently, series-parallel matrices are representation matrices of graphic matroids of series-parallel graphs, which can be recognized in linear time. We propose an algorithm that, for an m-by-n matrix A with k nonzeros, determines in expected time whether A is series-parallel or returns a minimal non–series-parallel submatrix of A. We complement the developed algorithm by an efficient [Formula: see text]implementation and report about computational results. History: Accepted by Andrea Lodi, Area Editor for Design &amp; Analysis of Algorithms–Discrete. Funding: This work was supported by Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant OCENW.M20.151]. Supplemental Material: The software that supports the findings of this study is available at the GitHub software repository ( https://github.com/discopt/cmr-series-parallel ).

Bonds Intersecting Long Paths in \(k\) -Connected Graphs

In 1966, Gallai asked whether there is a vertex which passes through all longest paths of a connected graph. Although this has been verified for some special classes of graphs such as outerplanar graphs, circular arc graphs, and series-parallel graphs, the answer is negative for general graphs. In this paper, we prove among other results that if we replace the vertex by a bond, then the answer is affirmative. A bond of a graph is a minimal nonempty edge-cut. In particular, in any 2-connected graph, the set of all edges incident to a vertex is a bond, called a vertex-bond. Clearly, for a 2-connected graph, a path passes through a vertex if and only if it meets the vertex-bond with respect to . Therefore, a very natural approach to Gallai’s question is to study whether there is a bond meeting all longest paths. Let denote the length of a longest path of connected graphs. We show that there is a bond meeting all paths of length at least and for any 2- and 3-connected graph, respectively. For any -connected graph , we show that there is a bond meeting all paths of length at least , where if is even and if is odd. Our results also provide analogs of the results on bonds meeting long cycles given in [P.-L. Wu, Combin. Probab. Comput., 6 (1997), pp. 107–113; and S. McGuinness, Combinatorica, 25 (2005), pp. 439–450].

Computing Bend-Minimum Orthogonal Drawings of Plane Series–Parallel Graphs in Linear Time

AbstractA planar orthogonal drawing of a planar 4-graph G (i.e., a planar graph with vertex-degree at most four) is a crossing-free drawing that maps each vertex of G to a distinct point of the plane and each edge of G to a polygonal chain consisting of horizontal and vertical segments. A longstanding open question in Graph Drawing, dating back over 30 years, is whether there exists a linear-time algorithm to compute an orthogonal drawing of a plane 4-graph with the minimum number of bends. The term “plane” indicates that the input graph comes together with a planar embedding, which must be preserved by the drawing (i.e., the drawing must have the same set of faces as the input graph). In this paper we positively answer the question above for the widely-studied class of series–parallel graphs. Our linear-time algorithm is based on a characterization of the planar series–parallel graphs that admit an orthogonal drawing without bends. This characterization is given in terms of the orthogonal spirality that each type of triconnected component of the graph can take; the orthogonal spirality of a component measures how much that component is “rolled-up” in an orthogonal drawing of the graph.

A Robust Theory of Series Parallel Graphs

Motivated by distributed data processing applications, we introduce a class of labeled directed acyclic graphs constructed using sequential and parallel composition operations, and study automata and logics over them. We show that deterministic and non-deterministic acceptors over such graphs have the same expressive power, which can be equivalently characterized by Monadic Second-Order logic and the graded µ-calculus. We establish closure under composition operations and decision procedures for membership, emptiness, and inclusion. A key feature of our graphs, called synchronized series-parallel graphs (SSPG), is that parallel composition introduces a synchronization edge from the newly introduced source vertex to the sink. The transfer of information enabled by such edges is crucial to the determinization construction, which would not be possible for the traditional definition of series-parallel graphs. SSPGs allow both ordered ranked parallelism and unordered unranked parallelism. The latter feature means that in the corresponding automata, the transition function needs to account for an arbitrary number of predecessors by counting each type of state only up to a specified constant, thus leading to a notion of counting complexity that is distinct from the classical notion of state complexity. The determinization construction translates a nondeterministic automaton with n states and k counting complexity to a deterministic automaton with 2 n 2 states and kn counting complexity, and both these bounds are shown to be tight. Furthermore, for nondeterministic automata a bound of 2 on counting complexity suffices without loss of expressiveness.

Decremental Optimization of Vertex-Coloring Under the Reconfiguration Framework

Suppose that we are given a positive integer k, and a k-(vertex-)coloring of a given graph G. Then we are asked to find a coloring of G using the minimum number of colors among colorings that are reachable from by iteratively changing a color assignment of exactly one vertex while maintaining the property of k-colorings. In this paper, we give linear-time algorithms to solve the problem for graphs of degeneracy at most two and for the case where . These results imply linear-time algorithms for series-parallel graphs and grid graphs. In addition, we give linear-time algorithms for chordal graphs and cographs. On the other hand, we show that, for any , this problem remains NP-hard for planar graphs with degeneracy three and maximum degree four. Thus, we obtain a complexity dichotomy for this problem with respect to the degeneracy of a graph and the number k of colors.

A linear-time certifying algorithm for recognizing generalized series–parallel graphs

The problems of recognizing series–parallel graphs, outerplanar graphs, and generalized series–parallel graphs have been studied separately in the past. Efficient algorithms have been presented. However, none of the algorithms are certifying. A certifying algorithm generates, in addition to its answer, a certificate that can be used by a checker (a separate algorithm) to verify the correctness of the answer. The certificate is positive if the answer is ‘yes’, and is negative if the answer is ‘no’. In this paper, an O ( | E | + | V | ) -time certifying algorithm that simultaneously determines if a graph G = ( V , E ) is series–parallel, outerplanar, or generalized series–parallel is presented. The positive certificates are a construction sequence for constructing G if G is series–parallel, a generalized construction sequence for constructing G if G is generalized series–parallel but not series–parallel, and the edge set of the exterior boundary of an outerplanar embedding of G if G is outerplanar. The negative certificates are forbidden subgraphs or forbidden structures of G . All these certificates are generated by making only one pass over G after a preprocessing step decomposing G into its biconnected components.

Weighted Modulo Orientations of Graphs and Signed Graphs

Given a graph $G$ and an odd prime $p$, for a mapping $f: E(G) \to {\mathbb Z}_p\setminus\{0\}$ and a ${\mathbb Z}_p$-boundary $b$ of $G$, an orientation $\tau$ is called an $(f,b;p)$-orientation if the net out $f$-flow is the same as $b(v)$ in ${\mathbb Z}_p$ at each vertex $v\in V(G)$ under orientation $D$. This concept was introduced by Esperet et al. (2018), generalizing mod $p$-orientations and closely related to Tutte's nowhere zero 3-flow conjecture. They proved that $(6p^2 - 14p + 8)$-edge-connected graphs have all possible $(f,b;p)$-orientations. In this paper, the framework of such orientations is extended to signed graph through additive bases. We also study the $(f,b;p)$-orientation problem for some (signed) graphs families including complete graphs, chordal graphs, series-parallel graphs and bipartite graphs, indicating that much lower edge-connectivity bound still guarantees the existence of such orientations for those graph families.

On computing the Hamiltonian index of graphs

For an integer r≥0 the r-th iterated line graphLr(G) of a graph G is defined by: (i) L0(G)=G and (ii) Lr(G)=L(L(r−1)(G)) for r>0, where L(G) denotes the line graph of G. The Hamiltonian Indexh(G) of G is the smallest r such that Lr(G) has a Hamiltonian cycle [Chartrand, 1968]. Checking if h(G)=k is NP-hard for any fixed integer k≥0 even for subcubic graphs G [Ryjáček et al., 2011]. We study the parameterized complexity of this problem with the parameter treewidth, tw(G), and show that we can find h(G) in time1O⋆((1+2(ω+3))tw(G)) where ω is the matrix multiplication exponent. Prior work on computing h(G) includes various O⋆(2O(tw(G)))-time algorithms for checking if h(G)=0 holds; i.e., whether G has a Hamiltonian Cycle [Cygan et al., FOCS 2011; Bodlaender et al., Inform. Comput., 2015; Fomin et al., JACM 2016]; an O⋆(tw(G)O(tw(G)))-time algorithm for checking if h(G)=1 holds; i.e., whether L(G) has a Hamiltonian Cycle [Lampis et al., Discrete Appl. Math., 2017]; and, most recently, an O⋆((1+2(ω+3))tw(G))-time algorithm for checking if h(G)=1 holds [Misra et al., CSR 2019]. Our algorithm for computing h(G) generalizes these results.The NP-hard Eulerian Steiner Subgraph problem takes as input a graph G and a specified subset K of terminal vertices of G and asks if G has an Eulerian2 subgraph H containing all the terminals. A key ingredient of our algorithm for finding h(G) is an algorithm which solves Eulerian Steiner Subgraph in O⋆((1+2(ω+3))tw(G)) time. To the best of our knowledge this is the first FPT algorithm for Eulerian Steiner Subgraph. Prior work on the special case of finding a spanning Eulerian subgraph (i.e., with K=V(G)) includes a polynomial-time algorithm for series-parallel graphs [Richey et al., 1985] and an O⋆(2O(n))-time algorithm for planar graphs on n vertices [Sau and Thilikos, 2010]. Our algorithm for Eulerian Steiner Subgraph generalizes both these results.

Perfect Italian domination in graphs: Complexity and algorithms

An Italian dominating function on a simple undirected graph G is a function f : V ( G ) ⟶ { 0 , 1 , 2 } satisfying the condition that for each vertex v with f ( v ) = 0 , ∑ u ∈ N G ( v ) f ( u ) ≥ 2 . An Italian dominating function f on G is called a perfect Italian dominating function on G if for each vertex v with f ( v ) = 0 , ∑ u ∈ N G ( v ) f ( u ) = 2 . The weight of a function f on a graph G , denoted by w ( f ) , is the sum ∑ v ∈ V ( G ) f ( v ) . For a simple undirected graph G , Min-PIDF is the problem of finding the minimum weight of a perfect Italian dominating function on G . First, we discuss the complexity difference between Min-PIDF and the problem of finding the minimum weight of an Italian dominating function. We then establish the NP -completeness of the decision version of Min-PIDF in chordal graphs and investigate the hardness of approximation of Min-PIDF in general graphs. Finally, we present linear time algorithms for computing the minimum weight of a perfect Italian dominating function in block graphs and series-parallel graphs.

Finding geodesics on graphs using reinforcement learning

It is well known in biology that ants are able to find shortest paths between their nest and the food by successive random explorations, without any mean of communication other than the pheromones they leave behind them. This striking phenomenon has been observed experimentally and modelled by different mean-field reinforcement-learning models in the biology literature. In this paper, we introduce the first probabilistic reinforcement-learning model for this phenomenon. In this model, the ants explore a finite graph in which two nodes are distinguished as the nest and the source of food. The ants perform successive random walks on this graph, starting from the nest and stopping when they first reach the food; the transition probabilities of each random walk depend on the realizations of all previous walks through some dynamic weighting of the graph. We discuss different variants of this model based on different reinforcement rules and show that slight changes in this reinforcement rule can lead to drastically different outcomes. We prove that the ants indeed eventually find the shortest path(s) between their nest and the food in two variants of this model and when the underlying graph is, respectively, any series-parallel graph and a five-edge nonseries-parallel losange graph. Both proofs rely on the electrical network method for random walks on weighted graphs and on Rubin’s embedding in continuous time. The proof in the series-parallel cases uses the recursive nature of this family of graphs, while the proof in the seemingly simpler losange case turns out to be quite intricate: it relies on a fine analysis of some stochastic approximation, and on various couplings with standard and generalised Pólya urns.