Triconnected Components Research Articles

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.

Convex Grid Drawings of Plane Graphs with Pentagonal Contours on <i>O</i>(<i>n</i><sup>2</sup>) Grids

In a convex grid drawing of a plane graph, all edges are drawn as straight-line segments without any edge-intersection, all vertices are put on grid points and all facial cycles are drawn as convex polygons. A plane graph G has a convex drawing if and only if G is internally triconnected, and an internally triconnected plane graph G has a convex grid drawing on an (n-1)×(n-1) grid if either G is triconnected or the triconnected component decomposition tree T(G) of G has two or three leaves, where n is the number of vertices in G. An internally triconnected plane graph G has a convex grid drawing on a 2n×2n grid if T(G) has exactly four leaves. Furthermore, an internally triconnected plane graph G has a convex grid drawing on a 6n×n2 grid if T(G) has exactly five leaves. In this paper, we show that an internally triconnected plane graph G has a convex grid drawing on a 20n×16n grid if T(G) has exactly five leaves. We also present an algorithm to find such a drawing in linear time. This is the first algorithm that finds a convex grid drawing of such a plane graph G in a grid of O(n2) size.

An alternative method for refined process structure trees (RPST)

PurposeThe refined process structure tree (RPST), the hierarchy of non-overlapping single-entry single-exit (SESE) regions of a process model, has been utilized for better comprehension and more efficient analysis of business process models. Existing RPST methods, based on the triconnected components of edges, fail to identify a certain type of SESE region. The purpose of this paper is to introduce an alternative method for generating a complete RPST utilizing rather simple techniques.Design/methodology/approachThe proposed method first focuses on the SESE regions of bonds and rigids, from the innermost ones to the outermost ones, utilizing dominance and post-dominance relations. Then, any SESE region of a series nested in a bond or a rigid is identified with a depth-first search variation. Two-phase algorithms and their completeness proofs, a software tool incorporating visualization of stepwise outcomes, and the experimental results of the proposed method are provided.FindingsThe proposed method utilizes simple techniques that allow their straightforward implementation. Visualization of stepwise outcomes helps process analysts to understand the proposed method and the SESE regions. Experiments with 604 SAP reference models demonstrated the limitation of the existing RPST methods. The proposed method, however, completely identified all types of SESE regions, defined with nodes, in less computation time than with the old methods.Originality/valueEach triconnected component of the undirected version of a process model is associated with a pair of boundary nodes without discriminating between the entry and the exit. Here, each non-atomic SESE region is associated with two distinct entry and exit nodes from the original model in the form of a directed graph. By specifying the properties of SESE regions in more comprehensible ways, this paper facilitates a deeper understanding of SESE regions rather than relying on the resulting RPST.

Faster algorithms for shortest path and network flow based on graph decomposition

We propose faster algorithms for the maximum flow problem and shortest path problems based on graph decomposition. Our algorithms first construct indices (data structures) from a given graph, then use them for solving the problems. Time complexities of our algorithms depend on the size of the maximum triconnected component in the graph, say $r$. Max flow indexing problem is a basic network flow problem, which consists of two phases. In a preprocessing phase we construct an index and in a query phase we process the query using the index. We can solve all pairs maximum flow problem and minimum cut problem using the indices. Our algorithms run faster than known algorithms if $r$ is small. The maximum flow problem can be solved in $\mathcal{O}(nr)$ time, which is faster than the best known $\mathcal{O}(nm)$ algorithm [Orlin, STOC 2013] if $r = o(m)$, where $n$ and $m$ are the numbers of vertices and edges in the given network, respectively. Distance oracle problem is a basic problem in shortest path, consisting of two phases. In preprocessing phase we construct index and in query phase we use the index to find shortest path between two vertices. We use these indices to solve single source shortest path and all pair shortest path problems. If the given graph is undirected and all the weights are non-negative integers, then our algorithm finds shortest path between two vertices in $\mathcal{O}(m)$ time. If the given graph is directed or the weights are non-negative real numbers then our algorithm finds shortest path between two vertices in $\mathcal{O}(m + n \log r)$ time. If the edge weights are real numbers (i.e some of the weights are negative) then our algorithm finds shortest path between two vertices in $\mathcal{O}(m + nr)$ time.

CONVEX DRAWINGS OF INTERNALLY TRICONNECTED PLANE GRAPHS ON O(n2) GRIDS

In a convex grid drawing of a plane graph, every edge is drawn as a straight-line segment without any edge-intersection, every vertex is located at a grid point, and every facial cycle is drawn as a convex polygon. A plane graph G has a convex drawing if and only if G is internally triconnected. It has been known that an internally triconnected plane graph G of n vertices has a convex grid drawing on a grid of O(n3) area if the triconnected component decomposition tree of G has at most four leaves. In this paper, we improve the area bound O(n3) to O(n2), which is optimal up to a constant factor. More precisely, we show that G has a convex grid drawing on a 2n × 4n grid. We also present an algorithm to find such a drawing in linear time.

Upward Spirality and Upward Planarity Testing

A digraph is upward planar if it admits a planar drawing where all edges are monotone in the upward direction. It is known that the problem of testing a digraph for upward planarity is NP-complete in general. This paper describes an $O(n^4)$-time upward planarity testing algorithm for all digraphs that have a series-parallel structure, where $n$ is the number of vertices of the input. This significantly enlarges the family of digraphs for which a polynomial-time testing algorithm is known. Furthermore, the study is extended to general digraphs, and a fixed parameter tractable algorithm for upward planarity testing is described, whose time complexity is $O(d^t \cdot t \cdot n^3 + d \cdot t^2 \cdot n + d^2 \cdot n^2)$ where $t$ is the number of triconnected components of the digraph and $d$ is an upper bound on the diameter of any split component of the digraph. Our results use the new notion of upward spirality that, informally speaking, is a measure of the “level of winding” that a triconnected component of a digraph $G$ can have in an upward planar drawing of $G$.

The refined process structure tree

We consider a workflow graph as a model for the control flow of a business process and study the problem of workflow graph parsing, i.e., finding the structure of a workflow graph. More precisely, we want to find a decomposition of a workflow graph into a hierarchy of sub-workflows that are subgraphs with a single entry and a single exit of control. Such a decomposition is the crucial step, for example, to translate a process modeled in a graph-based language such as BPMN into a process modeled in a block-based language such as BPEL. For this and other applications, it is desirable that the decomposition be unique, modular and as fine as possible, where modular means that a local change of the workflow graph can only cause a local change of the decomposition. In this paper, we provide a decomposition that is unique, modular and finer than in previous work. We call it the refined process structure tree. It is based on and extends similar work for sequential programs by Tarjan and Valdes [ACM POPL ’80, 1980, pp. 95–105]. We give two independent characterizations of the refined process structure tree which we prove to be equivalent: (1) a simple descriptive characterization that justifies our particular choice of the decomposition and (2) a constructive characterization that allows us to compute the decomposition in linear time. The latter is based on the tree of triconnected components (elsewhere also known as the SPQR tree) of a biconnected graph.

Non-planar core reduction of graphs

We present a reduction method that reduces a graph to a smaller core graph which behaves invariant with respect to non-planarity measures like crossing number, skewness, coarseness, and thickness. The core reduction is based on the decomposition of a graph into its triconnected components and can be computed in linear time. It has applications in heuristic and exact optimization algorithms for the non-planarity measures mentioned above. Experimental results show that this strategy reduces the number of edges by 45% in average for a widely used benchmark set of graphs.

CONVEX DRAWINGS OF PLANE GRAPHS OF MINIMUM OUTER APICES

In a convex drawing of a plane graph G, every facial cycle of G is drawn as a convex polygon. A polygon for the outer facial cycle is called an outer convex polygon. A necessary and sufficient condition for a plane graph G to have a convex drawing is known. However, it has not been known how many apices of an outer convex polygon are necessary for G to have a convex drawing. In this paper, we show that the minimum number of apices of an outer convex polygon necessary for G to have a convex drawing is, in effect, equal to the number of leaves in a triconnected component decomposition tree of a new graph constructed from G, and that a convex drawing of G having the minimum number of apices can be found in linear time.

TWO FIXED-PARAMETER TRACTABLE ALGORITHMS FOR TESTING UPWARD PLANARITY

In this paper we consider the problem of testing an arbitrary digraph G = (V,E) for upward planarity. In particular we describe two fixed-parameter tractable algorithms for testing the upward planarity of G. The first algorithm that we present can test the upward planarity of G in O(2t · t! · |V|2)-time where t is the number of triconnected components of G. The second algorithm that we present uses a standard technique known as kernelisation and can test the upward planarity of G in O(|V|2 + k4 · (2k)!)-time, where k = |E| - |V|.

Acyclically pushable bipartite permutation digraphs: An algorithm

Given a digraph D = ( V , A ) and an X ⊆ V , D X denotes the digraph obtained from D by reversing those arcs with exactly one end in X. A digraph D is called acyclically pushable when there exists an X ⊆ V such that D X is acyclic. Huang, MacGillivray and Yeo have recently characterized, in terms of two excluded induced subgraphs on 7 and 8 nodes, those bipartite permutation digraphs which are acyclically pushable. We give an algorithmic proof of their result. Our proof delivers an O ( m 2 ) time algorithm to decide whether a bipartite permutation digraph is acyclically pushable and, if yes, to find a set X such that D X is acyclic. (Huang, MacGillivray and Yeo's result clearly implies an O ( n 8 ) time algorithm to decide but the polynomiality of constructing X was still open.) We define a strongly acyclic digraph as a digraph D such that D X is acyclic for every X. We show how a result of Conforti et al [Balanced cycles and holes in bipartite graphs, Discrete Math. 199 (1–3) (1999) 27–33] can be essentially regarded as a characterization of strongly acyclic digraphs and also provides linear time algorithms to find a strongly acyclic orientation of an undirected graph, if one exists. Besides revealing this connection, we add simplicity to the structural and algorithmic results first given in Conforti et al [Balanced cycles and holes in bipartite graphs, Discrete Math. 199 (1–3) (1999) 27–33]. In particular, we avoid decomposing the graph into triconnected components. We give an alternate proof of a theorem of Huang, MacGillivray and Wood characterizing acyclically pushable bipartite tournaments. Our proof leads to a linear time algorithm which, given a bipartite tournament as input, either returns a set X such that D X is acyclic or a proof that D is not acyclically pushable.

Solid Model Reconstruction of Wireframe CAD Models Based on Topological Embeddings of Planar Graphs

This paper describes a robust and versatile method based on graph embedding for reconstructing a solid model from a wireframe model. The robustness and versatility of the conventional methods are limited in that: (1) most of them are heuristic and thus less robust, and (2) the rest, deterministic ones, can handle only small class of wireframes. Unlike the conventional methods, our approach is deterministic and covers a larger class of wireframes that are topologically 2-connected planar multigraphs. The class includes wireframes that can be interpreted as a closed two-manifold in multiple ways. The proposed algorithm consists of three steps: (1) all topological solutions are exhaustively generated using triconnected component decomposition; (2) the surface geometries for all the topological solutions are generated; and (3) the solutions are pruned down to geometrically valid ones. We also show the algorithm is extendable to the class of general planar multigraphs. The approach is characterized by generating the complete set of topological solutions without referring to the geometry of the wireframe, which makes the process free from geometric errors and instabilities. The algorithm is also fast, because even when there are many topological solutions, the total number of different faces is very small. The proposed approach provides a method for easy and intuitive geometric modeling as well as a conversion tool for legacy wireframes.

Revisiting decomposition analysis of geometric constraint graphs

Geometric problems defined by constraints can be represented by geometric constraint graphs whose nodes are geometric elements and whose arcs represent geometric constraints. Reduction and decomposition are techniques commonly used to analyze geometric constraint graphs in geometric constraint solving. In this paper we first introduce the concept of deficit of a constraint graph. Then we give a new formalization of the decomposition algorithm due to Owen. This new formalization is based on preserving the deficit rather than on computing triconnected components of the graph and is simpler. Finally we apply tree decompositions to prove that the class of problems solved by the formalizations studied here and other formalizations reported in the literature is the same.

Incremental Convex Planarity Testing

An important class of planar straight-line drawings of graphs are convex drawings, in which all the faces are drawn as convex polygons. A planar graph is said to be convex planar if it admits a convex drawing. We give a new combinatorial characterization of convex planar graphs based on the decomposition of a biconnected graph into its triconnected components. We then consider the problem of testing convex planarity in an incremental environment, where a biconnected planar graph is subject to on-line insertions of vertices and edges. We present a data structure for the on-line incremental convex planarity testing problem with the following performance, where n denotes the current number of vertices of the graph: (strictly) convex planarity testing takes O(1) worst-case time, insertion of vertices takes O(log n) worst-case time, insertion of edges takes O(log n) amortized time, and the space requirement of the data structure is O(n).

Fully dynamic planarity testing with applications

This paper introduces compressed certificates for planarity, biconnectivity and triconnectivity in planar graphs, and proves many structural properties of certificates in planar graphs. As an application of our compressed certificates, we develop efficient dynamic planar algorithms. In particular, we consider the following three operations on a planar graph G : (i) insert an edge if the resultant graph remains planar; (ii) delete an edge; and (iii) test whether an edge could be added to the graph without violating planarity. We show how to support each of the above operations in O ( n 2/3 ) time, where n is the number of vertices in the graph. The bound for tests and deletions is worst-case, while the bound for insertions is amortized. This is the first algorithm for this problem with sub-linear running time, and it affirmatively answers a question posed in Epstein et al. [1992]. We use our compressed certificates for biconnectivity and triconnectivity to maintain the biconnected and triconnected components of a dynamic planar graph. The time bounds are the same: O ( n 2/3 ) worst-case time per edge deletion, O ( n 2/3 ) amortized time per edge insertion, and O ( n 2/3 ) amortized time per edge insertion, and O ( n 2/3 )worst-case time to check whether two vertices are either biconnected or triconnected.

On-line maintenance of triconnected components with SPQR-trees

We consider the problem of maintaining on-line the triconnected components of a graphG. Letn be the current number of vertices ofG. We present anO(n)-space data structure that supports insertions of vertices and edges, and queries of the type Are there three vertex-disjoint paths between verticesv1 andv2? A sequence ofk operations takes timeO(k·ź(k, n)) ifG is biconnected(ź(k, n) denotes the well-known Ackermann's function inverse), and timeO(n logn+k) ifG is not biconnected. Note that the bounds do not depend on the number of edges ofG. We use theSPQR-tree, a versatile data structure that represents the decomposition of a biconnected graph with respect to its triconnected components, and theBC-tree, which represents the decomposition of a connected graph with respect to its biconnected components.

Upward Planar Drawing of Single-Source Acyclic Digraphs

An upward plane drawing of a directed acyclic graph is a plane drawing of the digraph in which each directed edge is represented as a curve monotone increasing in the vertical direction. Thomassen has given a nonalgorithmic, graph-theoretic characterization of those directed graphs with a single source that admit an upward plane drawing. This paper presents an efficient algorithm to test whether a given single-source acyclic digraph has an upward plane drawing and, if so, to find a representation of one such drawing. This result is made more significant in light of the recent proof by Garg and Tamassia that the problem is NP-complete for general digraphs. The algorithm decomposes the digraph into biconnected and triconnected components and defines conditions for merging the components into an upward plane drawing of the original digraph. To handle the triconnected components, we provide a linear algorithm to test whether a given plane drawing of a single-source digraph admits an upward plane drawing with the same faces and outer face, which also gives a simpler, algorithmic proof of Thomassen’s result. The entire testing algorithm (for general single-source directed acyclic graphs) operates in $O(n^2 )$ time and $O(n)$ space (n being the number of vertices in the input digraph) and represents the first polynomial-time solution to the problem.

On-Line Maintenance of Triconnected Components with SPQR-Trees

On-Line Maintenance of Triconnected Components with SPQR-Trees

Graph-based extraction of two-connected morphological features from boundary representations

In this paper a feature extraction technique to isolate two-connected features from a boundary model is discussed. The technique involves identification of faces with multiple edge loops and faces sharing multiple edges as candidates for entrance faces of such features. It is observed that the two-connected features constitute tri connected components in the edge-face graphs of the boundary models. Heuristics are used to enhance an algorithm to decompose the edge-face graph of a boundary representation into its triconnected components. Pros and cons of this method are compared with the classic graph theoretic algorithm used to obtain triconnected components from a graph.

A parallel algorithm for finding a triconnected component separator with an application

A parallel algorithm for finding a triconnected component separator with an application