Maximal Plane Graph Research Articles

List coloring and diagonal coloring for plane graphs of diameter two

Let G be a maximal plane graph of diameter at most two. Let χl(G) and χd(G) denote the list chromatic number and the diagonal chromatic number of G, respectively. In this paper, we study the structure of G and show the following results: (1) χl(G) ≤ 4; and (2) χd(G) ≤ 9, and χd(G)=9 if and only if G is a specific graph on nine vertices.

A Structural Property of Trees with an Application to Vertex-Arboricity

We provide a structural property of trees, which is applied to show that if a plane graph G contains two edge-disjoint spanning trees, then its dual graph G⁎ has the vertex-arboricity at most 2. We also show that every maximal plane graph of order at least 4 contains two edge-disjoint spanning trees.

Constrained floorplans in 2D and 3D

A rectilinear dual of a plane graph refers to a partition of a rectangular area into nonoverlapping rectilinear polygonal modules, where each module corresponds to a vertex such that two modules have a side-contact iff their corresponding vertices are adjacent. An interesting observation is that most algorithms yielding low polygonal complexity rectilinear duals require the use of ⊤-shape modules or their extensions. To justify the intuition that ⊤-shape modules are more powerful than other 8-sided modules, we show the optimum polygonal complexity of ⊤-free rectilinear duals to be exactly 12. Our construction uses only monotone staircase modules. We continue this line of research by studying polygonal complexity and area-universality of rectilinear duals using modules of constrained shapes.Motivated by the design challenges in 3D IC technology, we also study the generalization of rectilinear duals in three dimensions by allowing each module to be an orthogonal polyhedron. Such a generalization to three dimensions opens the door for non-planar graphs to be accommodated in floorplan design. In particular, we prove that all chordal graphs admit 3D floorplans, which parallels the well-known result that all maximal plane graphs admit rectilinear duals.

Two counterexamples on completely independent spanning trees

For each k ≥ 2 , we construct a k -connected graph which does not contain two completely independent spanning trees. This disproves a conjecture of Hasunuma. Furthermore, we also give an example of a 3-connected maximal plane graph not containing two completely independent spanning trees.

Closed rectangle-of-influence drawings for irreducible triangulations

A ( weak) closed rectangle-of-influence (RI for short) drawing is a straight-line planar grid drawing in which there is no other vertex inside or on the boundary of the axis parallel rectangle defined by the two end vertices of any edge. Biedl et al. (1999) [1] showed that a plane graph G has a closed RI drawing, if and only if it has no filled 3 -cycle (a cycle of 3 vertices such that there is a vertex in the proper interior). They also showed that such a graph G has a closed RI drawing in an ( n − 1 ) × ( n − 1 ) grid, where n is the number of vertices in G. They raised an open question on whether this grid size bound can be improved (Biedl et al., 1999 [1]). Without loss of generality, we investigate maximal plane graphs admitting closed RI drawings in this paper. They are plane graphs with a quadrangular exterior face, triangular interior faces and no filled 3-cycles, known as irreducible triangulations (Fusy, 2009 [2]). In this paper, we present a linear time algorithm that computes closed RI drawings for irreducible triangulations. Given an arbitrary irreducible triangulation G with n vertices, our algorithm produces a closed RI drawing with size at most ( n − 3 ) × ( n − 3 ) ; and for a random irreducible triangulation, the expected grid size of the drawing is ( 22 n 27 + O ( n ) ) × ( 22 n 27 + O ( n ) ) . We then prove that for arbitrary n ⩾ 4 , there is an n-vertex irreducible triangulation, such that any of its closed RI drawing requires a grid of size ( n − 3 ) × ( n − 3 ) . Thus the grid size of the drawing produced by our algorithm is tight. This lower bound also answers the open question posed in Biedl et al. (1999) [1] negatively.

Coupled choosability of plane graphs

AbstractA plane graph G is coupled k‐choosable if, for any list assignment L satisfying $|{{L}}({{x}})|= {{k}}$ for every ${{x}}\in {{V}}({{G}})\cup {{F}}({{G}})$, there is a coloring that assigns to each vertex and each face a color from its list such that any two adjacent or incident elements receive distinct colors. We prove that every plane graph is coupled 7‐choosable. We further show that maximal plane graphs, ${{K}}_{{4}}$‐minor free graphs, and plane graphs with maximum degree at most three are coupled 6‐choosable. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 27–44, 2008

Acute triangles in triangulations on the plane with minimum degree at least 4

In this paper, we show that every maximal plane graph with minimum degree at least 4 and m finite faces other than an octahedron can be drawn in the plane so that at least ( m + 3 ) / 2 faces are acute triangles. Moreover, this bound is sharp.

A bijection between realizers of maximal plane graphs and pairs of non-crossing Dyck paths

Schnyder trees, or realizers, of maximal plane graphs, are widely used in the graph drawing domain. In this paper, a bijection between realizers and pairs of non-crossing Dyck paths is proposed. The transformation of a realizer into a pair of non-crossing Dyck paths and the opposite operation can be done in linear time. Applying this bijection, we enumerate the number of realizers of size n and we can efficiently generate all of them. Résumé Les arbres de Schnyder, ou réaliseurs, d’un graphe maximal planaire, sont largement répandus dans le domaine du dessin de graphe. Nous proposons ici, une bijection entre les réaliseurs et les paires de chemins de Dyck qui ne se coupent pas. La transformation d’un réaliseur en une paire de chemin de Dyck et son inverse se font en temps linéaire. Utilisant cette bijection, nous pouvons énumérer les réaliseurs de taille n et nous pouvons les générer exhaustivement de manière efficace.

Acute triangles in 4-connected maximal plane graphs

In this paper, we show that every 4-connected maximal plane graph with m finite faces other than the octahedron can be drawn in the plane so that at least ( m + 3 ) / 2 faces are acute triangles. Moreover, this bound is sharp.

On Triangle Contact Graphs

It is proved that any plane graph may be represented by a triangle contact system, that is a collection of triangular disks which are disjoint except at contact points, each contact point being a node of exactly one triangle. Representations using contacts of T-or Y-shaped objects follow. Moreover, there is a one-to-one mapping between all the triangular contact representations of a maximal plane graph and all its partitions into three Schnyder trees.

Wagner's theorem for Torus graphs

Wagner's theorem (any two maximal plane graphs having p vertices are equivalent under diagonal transformations) is extended to maximal torus graphs, graphs embedded in the torus with a maximal set of edges present. Thus any maximal torus graph having p vertices may be diagonally transformed into any other maximal torus graph having p vertices. As with Wagner's theorem, a normal form representing an intermediate stage in the above transformation is displayed. This result, along with Wagner's theorem, may make possible constructive characterizations of planar and toroidal graphs, through a wholly combinatorial definition of diagonal transformation.