Vertex Incident Research Articles

Height of minor faces in plane normal maps

The height of a face in a plane graph is the greatest degree of a vertex incident with this face. Under appropriate necessary conditions we prove that any plane normal map has a 3-face of height at most 20, or a 4-face of height at most 11, or else a 5-face of height at most 5. The bounds 20 and 5 are shown to be the best possible.

On the Complexity of Arrangements of Circles in the Plane

Continuing and extending the analysis in a previous paper [15], we establish several combinatorial results on the complexity of arrangements of circles in the plane. The main results are a collection of partial solutions to the conjecture that (a) any arrangement of unit circles with at least one intersecting pair has a vertex incident to at most three circles, and (b) any arrangement of circles of arbitrary radii with at least one intersecting pair has a vertex incident to at most three circles, under appropriate assumptions on the number of intersecting pairs of circles (see below for a more precise statement).

Grow & fold: compressing the connectivity of tetrahedral meshes

Standard representations of irregular finite element meshes combine vertex data (sample coordinates and node values) and connectivity (tetrahedron–vertex incidence). Connectivity specifies how the samples should be interpolated. It may be encoded as four vertex-references for each tetrahedron, which requires 128 m bits where m is the number of tetrahedra in the mesh. Our ‘Grow & Fold’ format reduces the connectivity storage down to 7 bits per tetrahedron: three of these are used to encode the presence of children in a tetrahedron spanning tree; the other four constrain sequences of ‘folding’ operations, so that they produce the connectivity graph of the original mesh. Additional bits must be used for each handle in the mesh and for each topological ‘lock’ in the tree. However, as our experiments with a prototype implementation show, the increase of the storage cost due to this extra information is typically no more than 1–2%. By storing vertex data in an order defined by the tree, we avoid the need to store tetrahedron–vertex references and facilitate variable length coding techniques for the vertex data. We provide the details of simple, loss-less compression and decompression algorithms and discuss a way of decreasing the storage cost to about 6 bits per tetrahedron.

On 4-Valent Symmetric Graphs

Let G act transitively on incident vertex, edge pairs of the connected 4-valent graph Γ. If a normal subgroup N does not give rise to a natural 4-valent quotient ΓN with G/N acting transitively on incident vertex, edge pairs, then either (a) N has just one or two orbits on vertices, or (b) N has r ⩾ 3 orbits on vertices and the natural quotient ΓN is a circuit Cr (Theorem 1.1). We give a complete classification of the graphs arising in (a) when the normal subgroup N is elementary abelian (Theorems 1.2 and 1.3). Case (b), which depends to some extent on case (a), is more technical and is studied in a subsequent paper.

On an integral representation for the genus series for $2$-cell embeddings

An integral representation for the genus series for maps on oriented surfaces is derived from the combinatorial axiomatisation of 2-cell embeddings in orientable surfaces. It is used to derive an explicit expression for the genus series for dipoles. The approach can be extended to vertex-regular maps in general and, in this way, may shed light on the genus series for quadrangulations. The integral representation is used in conjunction with an approach through the group algebra, CO,, of the symmetric group [11] to obtain a factorisation of certain Gaussian integrals. 1. A POWER SERIES REPRESENTATION FOR THE GENUS SERIES A map is a 2-cell embedding of a connected unlabelled graph ', with loops and multiple edges allowed, in a closed surface X, without boundary, which is assumed throughout to be oriented. The deletion of ' separates X into regions homeomorphic to open discs, called the faces of the map, and the number of edges bordering a face is called its degree. A map is rooted by distinguishing a mutually incident vertex, edge and face. The genus series for a class of maps is the formal generating series for the number of inequivalent maps with respect to genus, and the numbers of vertices, edges and faces. It is assumed hereinafter that maps are rooted. The general approach adopted here combines ideas of Jackson and Visentin [11] with those of 't Hooft [8] and Bessis, Itzykson and Zuber [3] who, in the above terminology, derived the genus series for diagrams akin to a class of maps by techniques from conformal field theory. Although [81 and [3] are important papers, they have remained largely inaccessible to combinatorialists because of their uncertainty about the automorphisms of these diagrams as combinatorial structures. In this paper, an explicit construction is given for an integral representation for the genus series for general maps directly from the combinatorial axiomatisation for embeddings on oriented surfaces. Moreover, we also develop methods which are extensible to vertex-regular maps (vertices have the same degree) and thence, by restriction, to quadrangulations (maps whose faces are bounded by four edges). This is done by examining dipoles (maps with two vertices) in detail. Although the argument is an algebraic one, based on the ring of formal power series, to assert that particular series belong to the ring, it is necessary to Received by the editors July 1, 1992 and, in revised form, August 15, 1993. 1991 Mathematics Subject Classification. Primary 05A1 5, 20C1 5; Secondary 57N37. (?) 1994 American Mathematical Society 0002-9947/94 $1.00 + S.25 per page

An upper bound for the total chromatic number

The total chromatic number,χ″(G), of a graphG, is defined to be the minimum number of colours needed to colour the vertices and edges of a graph in such a way that no adjacent vertices, no adjacent edges and no incident vertex and edge are given the same colour. This paper shows that\(\chi ''\left( G \right) \leqslant \chi '\left( G \right) + 2\sqrt { \chi \left( G \right)} \), whereχ(G) is the vertex chromatic number andχ′(G) is the edge chromatic number of the graph.

On eliminating vertices from a tree

The removal of any vertex incident with at least three edges in a tree network would cause a significant structural change in the tree. We discuss the need for such vertex removals and establish relationships between distances on a tree resulting from removing at least one vertex and those on the original tree.

Algorithms of an Adaptive Reorganization of a Robot's Knowledge and Decision-making Systems

The paper discusses a class of problems which could be modelled in the form of dymanic graphs. The algorithms for adaptive correction of vertex incidence and optimal weights matrices are proposed for robots to utilize quickly new information and to develop the optimal behaviour in finite time.

On the planarity of regular incidence sequences

A graph is said to realize a sequence of positive integers if the set of vertex incidence numbers of the graph equals the given sequence. The class of regular sequences which have a planar realization is completely determined.