4-connected Maximal Planar Graph Research Articles

On Longest Cycles in Essentially 4-connected Planar Graphs

A planar 3-connected graph G is essentially 4-connected if, for any 3-separator S of G, one component of the graph obtained from G by removing S is a single vertex. Jackson and Wormald proved that an essentially 4-connected planar graph on n vertices contains a cycle C such that |V(C)|≥2n+45. For a cubic essentially 4-connected planar graph G, Grünbaum with Malkevitch, and Zhang showed that G has a cycle on at least 34n vertices. In the present paper the result of Jackson and Wormald is improved. Moreover, new lower bounds on the length of a longest cycle of G are presented if G is an essentially 4-connected planar graph of maximum degree 4 or G is an essentially 4-connected maximal planar graph.

On purely tree-colorable planar graphs

A tree-k-coloring of a graph G is a k-coloring of G such that the subgraph induced by the union of any two color classes is a tree. G is purely tree-k-colorable if the chromatic number of G is k and any k-coloring of G is a tree-k-coloring. Xu [16] conjectured that there exist only two purely tree-4-colorable 4-connected maximal planar graphs. In this paper, we construct an infinite family of purely tree-colorable 4-connected maximal planar graphs, called dumbbell-maximal planar graphs, which disprove Xu's conjecture. Moreover, we give the enumeration of dumbbell-maximal planar graphs and propose a conjecture on such graphs. It turns out that the conjecture implies naturally the uniquely 4-colorable planar graph conjecture.

Computing Cartograms with Optimal Complexity

In a rectilinear dual of a planar graph vertices are represented by simple rectilinear polygons, while edges are represented by side-contact between the corresponding polygons. A rectilinear dual is called a cartogram if the area of each region is equal to a pre-specified weight. The complexity of a cartogram is determined by the maximum number of corners (or sides) required for any polygon. In a series of papers the polygonal complexity of such representations for maximal planar graphs has been reduced from the initial 40 to 34, then to 12 and very recently to the currently best known 10. Here we describe a construction with 8-sided polygons, which is optimal in terms of polygonal complexity as 8-sided polygons are sometimes necessary. Specifically, we show how to compute the combinatorial structure and how to refine it into an area-universal rectangular layout in linear time. The exact cartogram can be computed from the area-universal layout with numerical iteration, or can be approximated with a hill-climbing heuristic. We also describe an alternative construction of cartograms for Hamiltonian maximal planar graphs, which allows us to directly compute the cartograms in linear time. Moreover, we prove that even for Hamiltonian graphs 8-sided rectilinear polygons are necessary, by constructing a non-trivial lower bound example. The complexity of the cartograms can be reduced to 6 if the Hamiltonian path has the extra property that it is one-legged, as in outer-planar graphs. Thus, we have optimal representations (in terms of both polygonal complexity and running time) for Hamiltonian maximal planar and maximal outer-planar graphs. Finally we address the problem of constructing small-complexity cartograms for 4-connected graphs (which are Hamiltonian). We first disprove a conjecture, posed by two set of authors, that any 4-connected maximal planar graph has a one-legged Hamiltonian cycle, thereby invalidating an attempt to achieve a polygonal complexity 6 in cartograms for this graph class. We also prove that it is NP-hard to decide whether a given 4-connected plane graph admits a cartogram with respect to a given weight function.

Hamiltonian cycles through prescribed edges of 4-connected maximal planar graphs

In 1956, W.T. Tutte proved that every 4-connected planar graph is hamiltonian. Moreover, in 1997, D.P. Sanders extended this to the result that a 4-connected planar graph contains a hamiltonian cycle through any two of its edges. It is shown that Sanders’ result is best possible by constructing 4-connected maximal planar graphs with three edges a large distance apart such that any hamiltonian cycle misses one of them. If the maximal planar graph is 5-connected then such a construction is impossible.

4-connected maximal planar graphs are 4-ordered

It is shown that in a 4-connected maximal planar graph there is for any four vertices a, b, c and d, a cycle in the graph that contains the four vertices and visits them in the order a, b, c and d.

On topological aspects of orientations

We are concerned with two classes of planar graphs: maximal planar graphs (i.e. polyhedral graphs, triangulations) and maximal bipartite planar graphs (i.e. bipartite planar graphs with quadrilateral faces). For these graphs we consider constrained orientations with a constant indegree for the internal vertex set. We recall or prove new fundamental relations between these orientations, specific tree decompositions and bipolar orientations. In particular, these relations yield linear time computation algorithms. Using these orientations, we give a characterization of 4-connected maximal planar graphs and 3-connected planar graphs, which leads to simple linear time algorithms (de Fraysseix, Ossona de Mendez, Dagasthul Seminar Proceedings, Submitted for Publication).

On the tightness conditions for maximal planar graphs

We characterize the tight structure of a vertex-accumulation-free maximal planar graph with no separating triangles. Together with the result of Halin who gave an equivalent form for such graphs, this yields that a tight structure always exists in every 4-connected maximal planar graph with one end.

Outerplanar Partitions of Planar Graphs

Anouterplanargraph is one that can be embedded in the plane so that all of the vertices lie on one of the faces. We investigate a conjecture of Chartrand, Geller, and Hedetniemi, that every planar graph can be edge-partitioned into two outerplanar subgraphs. We refute the stronger statement that every planarly embedded graph can be edge-partitioned into two outerplanar subgraphs, one of which is outerplanarly embedded. We give a method that yields outerplanar partitions of certain graphs not covered by previous results. We formulate a conjecture about 4-connected maximal planar graphs that implies the original conjecture. Finally, we verify a weaker form of the conjecture in which outerplanar subgraphs are replaced by subgraphs with no homeomorphs ofK4.

One-way infinite hamiltonian paths in infinite maximal planar graphs

A one-way infinite Hamiltonian path is constructed in an infinite 4-connected VAP-free maximal planar graph containing one or two vertices of infinite degree. Combining this result and that of R. HALIN who investigated the structure of such graphs, we conclude that such a path always exists in every infinite 4-connected maximal planar graph with exactly one end, which is an extension of H. WHITNEY'S theorem to infinite graphs.

A linear algorithm for finding Hamiltonian cycles in 4-connected maximal planar graphs

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