Showing papers on "Quartic graph published in 1980"

Filling regions in binary raster images: A graph-theoretic approach

Abstract: Filling regions in raster images is the term given to the problem of extracting a connected region (that contains some preselected seed pixel) and filling it with some color. A connected region is formally defined as the collection of all pixels that are in the transitive closure of a pixel-connectivity operator that is applied to the seed. For example, the 4-pixel-connectivity operator selects all pixels that are (spatially) 4-connected to the pixel operand, and have the same color. This problem can be solved relatively easily if the fill color is distinguishable so that every pixel, once colored (i.e. flagged OLD) will not be considered again. In binary images, the fill “color” is usually a binary pattern and therefore, a “colored” pixel may still have its previous (e.g., black or white) color. This fact makes the problem non-trivial.In this paper, a region to be filled is represented as a connected directed a-cyclic planar graph in which nodes are regular regions (defined below) that are easy to handle. An arc connects a regular region to its neighbor below which shares a common horizontal sub-boundary. Based on this abstract representation of a region, a formulation of the filling problem is developed as a variant of graph traversing. The difficulties imposed by filling with a binary pattern, and the avoidance of an explicit description of the graph are explored and a solution is presented. This solution utilizes the frame buffer (that is used to display the image) for improved efficiency of a graph traversal algorithm.This method turns out to be similar to [Lieberman-78], that is shown here to be incorrect. The complexity of the new algorithm is 0(N*L+ N*Log N), compared to 0(N*L*Log N) there, where N is the number of nodes in the graph and L is the average composite degree of each node. A proof of correctness for the new algorithm is given too.

An upper bound on the length of a Hamiltonian walk of a maximal planar graph

Abstract: A Hamiltonian walk of a connected graph is a shortest closed walk that passes through every vertex at least once, and the length of a Hamiltonian walk is the total number of edges traversed by the walk. We show that every maximal planar graph with p(≥ 3) vertices has a Hamiltonian cycle or a Hamiltonian walk of length ≤ 3(p - 3)/2.

Basis order for bond-graph junction structures

Abstract: Junction structures are bond graph representations of the multiport topology of energy-based systems. The basis variable order is the number of independent effort variables and flow variables that must be specified for a particular topology. Two algebraic equations are developed that predict the effort-flow basis order for an arbitrary simple junction structure (SJS). These numbers are shown to be completely determined by the topological characteristics of the SJS. The results are extended to weighted junction structures.

An efficient algorithm to find a Hamiltonian circuit in a 4-connected maximal planar graph

Abstract: This paper describes an efficient algorithm to find a Hamiltonian circuit in an arbitrary 4-connected maximal planar graph. The algorithm is based on our simlplified version of Whitney's proof of his theorem: every 4-connected maximal planar graph has a Hamiltonian circuit.

Generating All Planar 0-, 1-, 2-, 3-Connected Graphs

Abstract: A graph grammar is given which generates all k- connected planar graphs for given k=0,1,2,3; n=# of vertices, m=# of edges, and (for k=0) p=# of components. The method is based on two elementary operations, namely (1) Joining two nonadjacent vertices by an edge, and (2) Replacing a vertex by two adjacent vertices. The operations are applied to plane embeddings, in such a way that the result is again a plane graph. All essentially different embeddings are generated.