4-connected Triangulations Research Articles

On well-covered triangulations: Part II

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. A planar (simple) graph in which each face is a triangle is called a triangulation. It was proved in an earlier paper [A. Finbow, B. Hartnell, R. Nowakowski, M. Plummer, On well-covered triangulations: Part I, Discrete Appl. Math., 132, 2004, 97–108] that there are no 5-connected planar well-covered triangulations. It is the aim of the present paper to completely determine the 4-connected well-covered triangulations containing two adjacent vertices of degree 4. In a subsequent paper [A. Finbow, B. Hartnell, R. Nowakowski, M. Plummer, On well-covered triangulations: Part III (submitted for publication)], we show that every 4-connected well-covered triangulation contains two adjacent vertices of degree 4 and hence complete the task of characterizing all 4-connected well-covered planar triangulations. There turn out to be only four such graphs. This stands in stark contrast to the fact that there are infinitely many 3-connected well-covered planar triangulations.

Generating 4-connected plane triangulations(Theory)

Generating 4-connected plane triangulations(Theory)

A PTAS for the sparsest 2-spanner of 4-connected planar triangulations

A t-spanner of an undirected, unweighted graph G is a spanning subgraph S of G with the added property that for every pair of vertices in G, the distance between them in S is at most t times the distance between them in G. We are interested in finding a sparsest t-spanner, i.e., a t-spanner with the minimum number of edges. In the general setting, this problem is known to be NP-hard for all t⩾2. For t⩾5, the problem remains NP-hard for planar graphs, whereas for t∈{2,3,4}, the complexity of this problem on planar graphs is still unknown. In this paper we present a polynomial time approximation scheme for the problem of finding a sparsest 2-spanner of a 4-connected planar triangulation.

The largest real zero of the chromatic polynomial

It is proved that if every subcontraction of a graph G contains a vertex with degree at most k, then the chromatic polynomial of G is positive throughout the interval ( k, ∞); K k+1 shows that this interval is the largest possible. It is conjectured that the largest real zero of the chromatic polynomial of a χ-chromatic planar graph is always less than χ. For χ = 2 and 3, constructions are given for maximal maximally-connected χ-chromatic planar graphs (i.e., 3-connected quadrangulations for χ = 2 and 4-connected triangulations for χ = 3) whose chromatic polynomials have real zeros arbitrarily close to (but less than) χ.

On the complexity of optimization problems for 3-dimensional convex polyhedra and decision trees

We show that several well-known optimization problems involving 3-dimensional convex polyhedra and decision trees are NP-hard or NP-complete. One of the techniques we employ is a linear-time method for realizing a planar 3-connected triangulation as a convex polyhedron, which may be of independent interest.

Triangulating with high connectivity

We consider the problem of triangulating a given point set, using straight-line edges, so that the resulting graph is “highly connected”. Since the resulting graph is planar, it can be at most 5-connected. Under the nondegeneracy assumption that no three points are collinear, we characterize the point sets with three vertices on the convex hull that admit 4-connected triangulations. More generally, we characterize the planar point sets that admit triangulations having neither chords nor complex (i.e., nonfacial) triangles. We also show that any planar point set can be augmented with at most two extra points to admit a 4-connected triangulation. All our proofs are constructive, and the resulting triangulations can be constructed in O( n log n) time. We conclude by stating several open problems. In particular, it is open whether a polynomial-time algorithm exists for determining whether a point set with no degeneracy restrictions and no restrictions on the number of extreme points admits a 4- or 5-connected triangulation.

Generating the 4-connected and strongly 4-connected triangulations on the torus and projective plane

The triangulations of the torus can be generated from a set of 21 minimal triangulations by vertex splitting. We show that if we never create a 3-valent vertex when we split them we generate the 4-connected triangulations. In addition if we never create two adjacent 4-valent vertexes then we generate the strongly 4-connected triangulations.

Connectivity of plane triangulations

This paper gives a topological characterization of the sets of articulation in planar graphs. It leads to linear algorithms for testing the 3-connectivity, 4-connectivity and 5-connectivity for plane triangulations (i.e., topological planar graphs such that all faces, except possibly the external face, are circuits of length 3). These algorithms remain optimal when they are extended in order to enumerate all the articulation k-sets of a k-connected triangulation. This study uses subgraph listing algorithms developed by Chiba and Nishizeki. It is related to the Hamiltonian circuit problem: since all 4-connected planar graphs are Hamiltonian, and there are linear algorithms for finding Hamiltonian circuits in such graphs, the 4-connectivity test means that there is a 2-step linear process for finding Hamiltonian circuits that is guaranteed to work for 4-connected plane triangulations.

Random Triangulations of the Plane

It is shown that random 2- and 3-connected triangulations (bicubic maps) with 2 n faces (vertices) almost certainly contain cn , c >0, copies of any particular 2- or 3-connected triangulation (bicubic map), respectively. Almost all 2- and 3-connected triangulations, and bicubic maps, with m vertices have longest path length less than cm , for some c < 1. If Barnette's conjecture that every 3-connected bicubic map is hamiltonian is false then almost all 3-connected bicubic maps are counterexamples to it.

Interval representations of planar graphs

We prove that every planar graph is the intersection graph of a collection of three-dimensional boxes, with intersections occuring only in the boundaries of the boxes. Furthermore, we characterize the graphs that have such representations (called strict representations) in the plane. These are precisely the proper subgraphs of 4-connected planar triangulations, which we characterize by forbidden sub-graphs. Finally, we strengthen a result of E. R. Scheinerman (“Intersection Classes and Multiple Intersection Parameters”, Ph. D. thesis, Princeton Univ., 1984) to show that every planar graph has a strict representation using at most two rectangles per vertex.

Lattice distortion, specific heat, and antiferromagnetic ordering in AuMn

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. A planar (simple) graph in which each face is a triangle is called a triangulation. It was proved in an earlier paper Finbow et al. (2004) [3] that there are no 5-connected planar well-covered triangulations, and in Finbow et al. (submitted for publication) [4] that there are exactly four 4-connected well-covered triangulations containing two adjacent vertices of degree 4. It is the aim of the present paper to complete the characterization of 4-connected well-covered triangulations by showing that each such graph contains two adjacent vertices of degree 4.