4-connected Triangulation Research Articles

$K_6$-Minors in Triangulations on the Klein Bottle

In this paper, we shall characterize triangulations on the Klein bottle without $K_6$-minors. Our characterization implies that every 5-connected triangulation on the Klein bottle has a $K_6$-minor. The connectivity “5" is best possible in a sense that there is a 4-connected triangulation on the Klein bottle without $K_6$-minors.

Looseness width of 5-connected triangulations on the torus

Abstract Let G be a triangulation on a closed surface and c : V ( G ) → { 1 , 2 , 3 , … , 3 + k } a color assignment of the vertices of G. Then a face uvw of G is said to be heterochromatic for c if its three corners u, v and w receive three distinct colors. Furthermore, G is said to be k-loosely tight if there is a heterochromatic face of G for any surjection c : V ( G ) → { 1 , 2 , 3 , … , 3 + k } . The looseness of G, denoted by ξ ( G ) , is defined as the minimum k such that G is k-loosely tight. We show that if G is 5-connected triangulation on the torus, then ξ ( G ) is independent of the embedding of G.

On the matching extendability of graphs in surfaces

A graph G with at least 2 n + 2 vertices is said to be n-extendable if every matching of size n in G extends to a perfect matching. It is shown that (1) if a graph is embedded on a surface of Euler characteristic χ, and the number of vertices in G is large enough, the graph is not 4-extendable; (2) given g > 0 , there are infinitely many graphs of orientable genus g which are 3-extendable, and given g ¯ ⩾ 2 , there are infinitely many graphs of non-orientable genus g ¯ which are 3-extendable; and (3) if G is a 5-connected triangulation with an even number of vertices which has genus g > 0 and sufficiently large representativity, then it is 2-extendable.

Canonical Ordering Trees and Their Applications in Graph Drawing

We study the properties of Schnyder’s realizers and canonical ordering trees of plane graphs. Based on these newly discovered properties, we obtain compact drawings of two styles for any plane graph G with n vertices. First, we show that G has a visibility representation with height at most ⌈ 15n/16 ⌉. This improves the previous best bound of (n - 1). Second, we show that every plane graph G has a straight-line grid embedding on an (n - δ0 - 1) × (n - δ0 - 1) grid, where δ0 is the number of cyclic faces of G with respect to its minimum realizer. This improves the previous best bound of (n - 1) × (n - 1). We also study the properties of the regular edge labeling of 4-connected plane triangulation. Based on these properties, we show that every such a graph has a canonical ordering tree with at most ⌈ (n + 1)/2 ⌉ leaves. This improves the previously known bound of ⌊ (2n + 1)/3 ⌋. We show that every 4-connected plane graph has a visibility representation with height at most ⌈ 3n/4 ⌉. All drawings discussed in this paper can be obtained in linear time.

Planar triangulations which quadrangulate other surfaces

We shall show that for any closed surface F 2 except the sphere, there is a triangulation on the sphere which can be embedded on F 2 as a quadrangulation, and characterize those 5-connected triangulations on the sphere that quadrangulate other closed surfaces, discussing double 4-cycle covers of triangulations.

A theorem on paths in locally planar triangulations

In this note, we show that every 5-connected triangulation in a surface with sufficiently large representativity is Hamiltonian-connected.

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 Size of the Largest Components in Random Planar Maps

Bender, Richmond, and Wormald showed that in almost all planar 3-connected triangulations (or dually, 3-connected cubic maps) with n edges, the largest 4-connected triangulation (or dually, the largest cyclically 4-edge-connected cubic component) has about n/2 edges [ Random Structures Algorithms, 7 (1995), pp. 273--285]. In this paper, we derive some general results about the size of the largest component and apply them to a variety of types of planar maps.

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.

Disjoint paths, planarizing cycles, and spanning walks

We study the existence of certain disjoint paths in planar graphs and generalize a theorem of Thomassen on planarizing cycles in surfaces. Results are used to prove that every 5-connected triangulation of a surface with sufficiently large representativity is hamiltonian, thus verifying a conjecture of Thomassen. We also obtain results about spanning walks in graphs embedded in a surface with large representativity.

An extension of whitney’s theorem to infinite strong triangulations

In this paper, the existence of one-way infinite Hamiltonian paths in an infinite 4-connected strong triangulation is constructively verified, which is an extension of WHITNEY’S theorem to infinite graphs.

Spanning Trees in Locally Planar Triangulations

Thomassen has recently shown that a triangulation of the orientable surface of genus g has a spanning tree of maximum degree at most 4, provided every noncontractible cycle has length at least 2 3 g + 4 . We show that a 4-connected triangulation of the orientable surface of genus g has a spanning tree of maximum degree at most 3, provided that every noncontractible cycle has length at least 2 3 g + 5 . This proves a result suggested by Thomassen. Examples demonstrate that some condition on the length of the noncontractible cycles is necessary for a result of this kind.