17799789
Cell rotation graphs of strongly connected orientations of plane graphs with an application
The cell rotation graph D(G) on the strongly connected orientations of a 2-edge-connected plane graph G is defined. It is shown that D(G) is a directed forest and every component is an in-tree with one root; if T is a component of D(G), the reversions of all orientations in T induce a component of D(G), denoted by T-, thus (T,T-) is called a pair of in-trees of D(G); G is Eulerian if and only if D(G) has an odd number of components (all Eulerian orientations of G induce the same component of D(G)); the width and height of T are equal to that of T-, respectively. Further it is shown that the pair of directed tree structures on the perfect matchings of a plane elementary bipartite graph G coincide with a pair of in-trees of D(G). Accordingly, such a pair of in-trees on the perfect matchings of any plane bipartite graph have the same width and height.
39584893
A solution to Gutman's problem on the characteristic polynomial of a bipartite graph
Abstract In this short paper, we present a solution to Gutman's problem on the characteristic polynomial of a bipartite graph (Research Problem 134, Discrete Math. 88 (1991)).
15887745
The Rotation Graphs of Perfect Matchings of Plane Bipartite Graphs
Abstract In the present paper, the minimal proper alternating cycle (MPAC) rotation graph R(G) of perfect matchings of a plane bipartite graph G is defined. We show that an MPAC rotation graph R(G) of G is a directed rooted tree, and thus extend such a result for generalized polyhex graphs to arbitrary plane bipartite graphs. As an immediate result, we describe a one-to-one correspondence between MPAC systems and perfect matchings in G.
29766053
Construction for bicritical graphs and k-extendable bipartite graphs
A graph G is said to be bicritical if G-u-v has a perfect matching for every choice of a pair of points u and v. Bicritical graphs play a central role in decomposition theory of elementary graphs with respect to perfect matchings. As Plummer pointed out many times, the structure of bicritical graphs is far from completely understood. This paper presents a concise structure characterization on bicritical graphs in terms of factor-critical graphs and transversals of hypergraphs. A connected graph G with at least 2k+2 points is said to be k-extendable if it contains a matching of k lines and every such matching is contained in a perfect matching. A structure characterization for k-extendable bipartite graphs is given in a recursive way. Furthermore, this paper presents an O(mn) algorithm for determining the extendability of a bipartite graph G, the maximum integer k such that G is k-extendable, where n is the number of points and m is the number of lines in G.
34749580
Plane elementary bipartite graphs
A connected graph is called elementary if the union of all perfect matchings forms a connected subgraph. In this paper we mainly study various properties of plane elementary bipartite graphs so that many important results previously obtained for hexagonal systems are treated in a unified way. Firstly, we show that a plane bipartite graph G is elementary if and only if the boundary of each face (including the infinite face) is an alternating cycle with respect to some perfect matching of G. For a plane bipartite graph G all interior vertices of which are of the same degree, a stronger result is obtained; namely, G is elementary if and only if the boundary of the infinite face of G is an alternating cycle with respect to some perfect matching of G. Second, the concept of the Z-transformation graph Z(G) of a hexagonal system G (whose vertices represent the perfect matchings of G) is extended to a plane bipartite graph G and some results analogous to those for hexagonal systems are obtained. A peripheral face f of G is called reducible if the removal of the internal vertices and edges of the path that is the intersection of f and the exterior face of G results in a plane elementary bipartite graph. Thirdly, we obtain the reducible face decomposition for plane elementary bipartite graphs. Furthermore, sharp upper and lower bounds for the number of reducible faces are derived. Conversely, we can construct any plane elementary bipartite graphs by adding new peripheral faces one by one. As applications of this approach, we give simple construction methods for several types of plane elementary bipartite graphs G that contain a forcing edge (which belongs to exactly one perfect matching of G) and whose Z-transformation graphs Z(G) contain vertices of degree one.
41201857
A Note on the Number of Perfect Matchings of Bipartite Graphs
Abstract Let G be a bipartite graph with 2n vertices, A its adjacency matrix and K the number of perfect matchings. For plane bipartite graphs each interior face of which is surrounded by a circuit of length 4s + 2, s ϵ {1, 2, …}, an elegant formula, i.e. det A = (−1)nK2, had been rigorously proved by Cvetkovic et al. (1982). For general bipartite graphs, this note contains a necessary and sufficient condition for the above relation to hold. A fast algorithm to check if a plane bipartite graph has such a relation is given.
1887353
An Application of Well-Orderly Trees in Graph Drawing
Well-orderly trees seem to have the potential of becoming a powerful technique capable of deriving new results in graph encoding, graph enumeration and graph generation [3, 4]. In this paper, we reduce the height of the visibility representation of plane graphs from 5n/6 to (4n–1)/5, by using well-orderly trees.
8788311
Factoring Synchronous Grammars by Sorting
Synchronous Context-Free Grammars (SCFGs) have been successfully exploited as translation models in machine translation applications. When parsing with an SCFG, computational complexity grows exponentially with the length of the rules, in the worst case. In this paper we examine the problem of factorizing each rule of an input SCFG to a generatively equivalent set of rules, each having the smallest possible length. Our algorithm works in time O(n log n), for each rule of length n. This improves upon previous results and solves an open problem about recognizing permutations that can be factored.
36919902
Motion planning with uncertainty
We present a general framework for motion planning of robots in the presence of obstacles and other robots. We use variational calculus and optimization to find optimal open loop and closed loop plans in the presence of uncertainty. The plans are based on world models with set-valued uncertainty associated with the positions and shape of the obstacles. The open loop plans are generated by an efficient method that allows successive refinements of a nominal motion plan and accommodates finer levels of granularity as additional information becomes available. The closed loop plans are control policies that are based on given sensor models. The optimal plan is a control policy that performs the best in the worst-case situation. We discuss how the open loop and closed loop plans can be viewed as optimal strategies in the framework of two-person, zero-sum, non-cooperative games.
8798967
Prototype Modeling from Sketched Silhouettes based on Convolution Surfaces
This paper presents a hybrid method for creating three-dimensional shapes by sketching silhouette curves. Given a silhouette curve, we approximate its medial axis as a set of line segments, and convolve a linearly weighted kernel along each segment. By summing the fields of all segments, an analytical convolution surface is obtained. The resulting generic shape has circular cross-section, but can be conveniently modified via sketched profile or shape parameters of a spatial transform. New components can be similarly designed by sketching on different projection planes. The convolution surface model lends itself to smooth merging between the overlapping components. Our method overcomes several limitations of previous sketched-based systems, including designing objects of arbitrary genus, objects with semi-sharp features, and the ability to easily generate variants of shapes.
39496103
An Application of Well-orderly Trees in Graph Drawing
Well-orderly tree is a powerful technique capable of deriving new results in graph encoding, graph enumeration and graph generation [3, 5]. In this paper, by using well-orderly trees, we prove that any plane graph G with n vertices has a visibility representation with height , which can be constructed in linear time. This improves the best previous bound of .