207679368
A Distributive Lattice on the Set of Perfect Matchings of a Plane Bipartite Graph
Let G be a plane bipartite graph and M(G) the set of perfect matchings of G. The Z-transformation graph of G is defined as a graph on M(G): M,M′∈M(G) are joined by an edge if and only if they differ only in one cycle that is the boundary of an inner face of G. A property that a certain orientation of the Z-transformation graph of G is acyclic implies a partially ordered relation on M(G). An equivalent definition of the poset M(G) is discussed in detail. If G is elementary, the following main results are obtained in this article: the poset M(G) is a finite distributive lattice, and its Hasse diagram is isomorphic to the Z-transformation digraph of G. Further, a distributive lattice structure is established on the set of perfect matchings of any plane bipartite graph.
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.
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.
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.
8279644
Z-transformation graphs of maximum matchings of plane bipartite graphs
Let G be a plane bipartite graph. The Z-transformation graph Z(G) and its orientation Z(G) on the maximum matchings of G are defined. If G has a perfect matching, Z(G) and Z(G) are the usual Z-transformation graph and digraph. If G has neither isolated vertices nor perfect matching, then Z(G) is not connected. This paper mainly shows that some basic results for Z-transformation graph (digraph) of a plane elementary bipartite graph still hold for every nontrivial component of Z(G) (Z(G)). In particular, by obtaining a result that every shortest path of Z(G) from a source of Z(G) corresponds to a directed path of Z(G), we show that every nontrivial component of Z(G) has exactly one source and one sink. Accordingly, it follows that the block graph of every nontrivial component of Z(G) is a path. In addition, we give a simple characterization for a maximum matching of G being a cut-vertex of Z(G).
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.
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 .
17695045
A Multi-Agent Approach for Peer-to-Peer Based Information Retrieval System
This paper develops and analyzes distributed search techniques for use in a peer-to-peer (P2P) network-based Information Retrieval (IR) system. In the absence of a centralized mediator with global knowledge that directs requests to appropriate agents, agents must cooperate to forward the queries among themselves so as to find appropriate agents, and return and merge the results in order to fulfill the information retrieval task in a distributed environment. In our approach, the agent society is connected through an agent-view structure maintained by each agent. Initially, the agent-view structures are formed by agents connecting to each other randomly. However, we show that such an approach can be significantly enhanced by dynamically reorganizing the underlying agent-view topology and deploying contextsensitive distributed search algorithms. Experimental results indicate that appropriate organizational structures and distributed search mechanisms can have a positive influence on system performance.
8107037
Forward Shadow Mapping
Forward shadow mapping is a new approach to real-time shadow generation. The traditional shadow map algorithm maps the pixels in the eye’s view backward into the depth buffers of light sources (i.e. shadow maps), which is similiar to and often implemented as an extension to texture mapping. Our algorithm reverses this process by using 3-D image warping techniques to transform shadow map pixels forward into the eye’s view to directly indicate which pixels are lit; it does not interfere with normal texture mapping and easily supports antialiased shadow edges and projective textures. Access to shadow maps and projective textures is in pixel-sequential order. This algorithm has advantages when speed of texture mapping becomes the performance bottleneck, which is often the case in visual simulation and game applications.
940067
Visibility representation of plane graphs via canonical ordering tree,
In a visibility representation (VR for short) of a plane graph G, each vertex of G is represented by a horizontal line segment such that the line segments representing any two adjacent vertices of G are joined by a vertical line segment. Rosenstiehl and Tarjan [Rectilinear planar layouts and bipolar orientations of planar graphs, Discrete Comput. Geom. 1 (1986) 343], Tamassia and Tollis [An unified approach to visibility representations of planar graphs, Discrete Comput. Geom. 1 (1986) 321] independently gave linear time VR algorithms for 2-connected plane graph. Afterwards, one of the main concerns for VR is the size of the representation. In this paper, we prove that any plane graph G has a VR with height bounded by ⌊5n/6⌋. This improves the previously known bound ⌊15n/16⌋. We also construct a plane graph G with n vertices where any VR of G requires a size of (⌊2n/3⌋) × (⌊4n/3⌋ - 3). Our result provides an answer to Kant's open question about whether there exists a plane graph G such that all of its VR require width greater that cn, where c > 1 [G. Kant, A more compact visibility representation, Internat. J. Comput. Geom. Appl. 7 (1997) 197].
6607794
Can an Overlay Compensate for a Careless Underlay?
We investigate the ability of an overlay network to compensate for “careless” routing in the native network layer, i.e., for network-layer routes not optimized for performance. We consider cost/traffic-dependent, overlay routing on top of an underlay routing based on randomly-generated, cost/traffic-independent link weights, and determine the extent to which overlay-overcareless-underlay can achieve performance close to that attainable when underlay routing is performed in an optimal cost/trafficdependent (“careful”) manner. We identify three graph-theoretic metrics that collectively characterize the richness of the underlay network: the tightness (characteristic path length, or CPL), the thickness (cut average) and the weighted sum of node degrees. We find that only when the underlay graph is rich can the overlay compensate for a careless underlay. If network links have linear costs, we prove that overlay cost is a linear increasing function of CPL under some assumptions. Finally, we investigate heuristics (based on the notion of richness) for choosing a set of overlay nodes of fixed size that is likely to result in good overlay performance.