Abstract: We propose a linear-time algorithm for generating a planar layout of a planar graph. Each vertex is represented by a horizontal line segment and each edge by a vertical line segment. All endpoints of the segments have integer coordinates. The total space occupied by the layout is at mostn by at most 2n---4. Our algorithm, a variant of one by Otten and van Wijk, generally produces a more compact layout than theirs and allows the dual of the graph to be laid out in an interlocking way. The algorithm is based on the concept of abipolar orientation. We discuss relationships among the bipolar orientations of a planar graph.

Abstract: We call a graph (m, k)-colorable if its vertices can be colored with m colors in such a way that each vertex is adjacent to at most k vertices of the same color as itself. For the class of planar graphs, and the class of outerplanar graphs, we determine all pairs (m, k) such that every graph in the class is (m, k)-colorable. We include an elementary proof (not assuming the truth of the four-color theorem) that every planar graph is (4, 1)-colorable. Finally, we prove that, for each compact surface S, there is an integer k = k(S) such that every graph in S can be (4, k)-colored; we conjecture that 4 can be replaced by 3 in this statement.

Abstract: We show that the problem of computing source-sink reliability is NP-hard, in fact # P-complete, even for undirected and acyclic directed source-sink planar graphs having vertex degree at most three. Thus the source-sink reliability problem is unlikely to have an efficient algorithm, even when the graph can be laid out on a rectilinear grid.

Abstract: This paper presents an unbounded-parallel algorithm for performing a depth-first search of a planar undirected graph. The algorithm uses $O(n^4 )$ processors and executes in $O(\log ^3 n)$-time. It had previously been conjectured that the problem of computing a depth-first spanning tree was inherently sequential.

Abstract: We develop an O(n) algorithm to construct a rectangular dual of an n-vertex planar triangulated graph.

Abstract: A routing problem is given by a planar graph G= (V, E) with a given embedding into the plane and a set Ne of nets. A net is a pair of points on the boundary of the infinite face. The goal is to find a set of pairwise edge-disjoint paths connecting the terminals of the various nets. We assume that the degree of every vertex not on the boundary of the infinite face is even and call such routing problems half-even. We show that one can decide in time O(bn) whether a half-even problem is solvable and that a solution can be constructed in time O(n 2). Here n=¦V¦ and b is the number of vertices on the boundary of the infinite face. If the routing problem is even, i.e. every cut has even free capacity, and G is a subgraph of the planar grid then a solution can be found in time O(n 3/2).

Abstract: Map grammars are discussed first of all in their relationships to graph grammars. Maps are more strictly specified and better implementable structures than planar graphs. While graphs consist of vertices and edges, maps can be defined as sets of vertices, edges and regions. After the original introduction of sequential map grammars, most of the recent work has been on parallel map generating systems. The main types of these are: (1) binary fission/fusion systems with labeling and interactions of the regions, (2) the map interpretations of parallel graph grammars (such as propagating graph OL-systems) with node (region) labeling, and (3) the edgelabel-controlled binary propagating map OL-systems (BPMOL-systems). Of the latter systems various classes are defined: with single or double edge labels, and with edge insertion controlled by cicular words or by markers. Properties and applicability to cell division patterns of these families of systems are compared.

Abstract: C. Thomassen extended Tutte's theorem on cycles in planar graphs in the paper “A Theorem on Paths in Planar Graphs”. This note corrects a flaw in his proof.

Abstract: We present a parallel algorithm for colouring outer-planar graphs with minimal possible number of colours (at most 3). The algorithm runs in O(log n) time, O(n) space and uses O(n) processors on a concurrent-read concurrent-write (CRCW) parallel random access machine, where n is the number of vertices of a given graph. As an application we obtain a six-colouring parallel algorithm for planar graphs running on the same model of parallel computations in O(log n) time, O(n3) space and using O(n4) processors.

Abstract: Definition de la notion de dependance des sous graphes d'un graphe planaire. Borne superieure du nombre de copies d'un graphe planaire 3-connexe dependantes

Abstract: This paper extends the author's parallel nested dissection algorithm of [PR] originally devised for solving sparse linear systems. We present a class of new applications of the nested dissection method, this time to path algebra computations, (in both cases of single source and all pair paths), where the path algebra problem is defined by a symmetric matrix A whose associated graph G with n vertices is planar. We substantially improve the known algorithms for path algebra problems of that general class: {fx470-1}

Abstract: We propose a linear-time algorithm for generating a planar layout of a planar graph Each vertex is represented by a horizontal line segment and each edge by a vertical line segment All endpoints of the segments have integer coordinates The total space occupied by the layout is at most n by at most 2n -4 Our algorithm, a variant of one by Otten and van Wijk, generally produces a more compact layout than theirs and allows the dual of the graph to be laid out in an interlocking way The algorithm is based on the concept of a bipolar orientation We discuss relation- ships among the bipolar orientations of a planar graph

Abstract: We show that the Min Cut Linear Arrangement Problem is NP-complete for trees with polynomial size weights and derive from this the NP-completeness of Min Cut for planar graphs with maximum vertex degree 3. This is used to show the NP-completeness of Search Number, Vertex Separation, Progressive Black/White Pebble Demand, and Topological Bandwidth for planar graphs with maximum vertex degree 3.

Abstract: A common structure arising in computational geometry is the subdivision of a plane defined by the faces of a straight line planar graph. We consider a natural generalization of this structure on a polyhedral surface. The regions of the subdivision are bounded by geodesics on the surface of the polyhedron. A method is given for representing such a subdivision that is efficient both with respect to space and the time required to answer a number of different queries involving the subdivision. For example, given a point

Abstract: We prove the following theorem. Let G = (V, E) be a planar bipartite graph, embedded in the euclidean plane. Let O and I be two of its faces. Then there exist pairwise edge-disjoint cuts C1, …, Ct so that for each two vertices v, w with v, w ϵ O of v, w ϵ I, the distance from v to w in G is equal to the number of cuts Cj separating v and w. This theorem is dual to a theorem of Okamura on plane multicommodity flows, in the same way as a theorem of Karzanov is dual to one of Lomonosov.

Abstract: A planar graph G = V, E is an Inner-Four-Cycle-Free graph IFCF-graph if, after all series and parallel replacements, it has a planar embedding G' with a face f such that every cycle of G' having four or more edges contains a vertex of f It is shown that an IFCF-graph can be reduced to an edge by a recursive application of series, parallel and Δ-Y replacements This property forms the basis of an O|V|2 algorithm to compute the probability that a probabilistic IFCF-graph is connected This result is extended to graphs whose triconnected components are IFCF such graphs are not necessarily IFCF The class of IFCF-graphs includes the series-parallel graphs, wheel-graphs, ladder networks and graphs in normal form, for which polynomial network reliability algorithms are already known

Abstract: We investigate visibility representations of planar graphs, which are constructed by mapping vertices to horizontal segments, and edges to vertical segments that intersect only adjacent vertex-segments. We consider three types of visibility representations, and present linear time algorithms for testing the existence of and constructing visibility representations. Applications of our results can be found in VLSI layout compaction, and in efficient embedding of graphs in the rectilinear grid.

Abstract: An algorithm is presented for reconstructing visible regions from visible edge segments in object space. This has applications in hidden surface algorithms operating on polyhedral scenes and in cartography. A special case of reconstruction can be formulated as a graph problem: “Determine the faces of a straight-edge planar graph given in terms of its edges.” This is accomplished inO(n logn) time using linear space for a graph withn edges, and is worst-case optimal. The graph may have separate components but the components must not contain each other. The general problem of reconstruction is then solved by applying our algorithm to each component in the containment relation.

Abstract: Outerplanar graphs have been recently generalized in many directions. Almost all generalizations have been introduced to parameterize the family of planar graphs so that in consequence some of the decision problems which are NP-complete for planar graphs and easy (or trivial) for outerplanar graphs can be solved in polynomial time for every fixed value of a parameter. In this paper, we survey some families of graphs which are known as generalizations of outerplanar graphs: tree-structured graphs (e.g., series-parallel, k-terminal and Halin graphs), W-outerplanar and k-outerplanar. We discuss also some problems which are formulated by slightly modified versions of the statements that characterize outerplanar graphs: the largest face problem and the (independent) face covers in plane graphs. Almost every (sub)section of the paper contains an open problem, a good starting point for further research. Our description of results is rather informal, the reader interested in details is referred to the original works.