We show that the game chromatic number of a planar graph is at most 33. More generally, there exists a function f: f\l --+ f\l so that for each n E f\l. if a graph does not contain a homeomorph of Kn• then its game chromatic number is at most f(n). In particular, the game chromatic number of a graph is bounded in terms of its genus. Our proof is motivated by the concept of p-arrangeability, whi...

This paper investigates the problem of embedding planar graphs in books of few pages. An efficient algorithm for embedding a planar graph in a book establishes an upper bound of seven pages for any planar graph. This disproves a conjecture of Bernhart and Kainen that the pagenumber of a planar graph can be arbitrarily large. I t is also shown that the stellations of K3 have pagenumber three, th...

The square G2 of a graph G is the graph with the same vertex set as G and with two vertices adjacent if their distance in G is at most 2. Thomassen showed that for a planar graph G with maximum degree ∆(G) = 3 we have χ(G2) ≤ 7. Kostochka and Woodall conjectured that for every graph, the list-chromatic number of G2 equals the chromatic number of G2, that is χl(G 2) = χ(G2) for all G. If true, t...

A natural way to draw two planar graphs whose vertex sets are matched is to assign each matched pair a unique y-coordinate. In this paper we introduce the concept of such matched drawings, which are a relaxation of simultaneous geometric embeddings with mapping. We study which classes of graphs allow matched drawings and show that (i) two 3-connected planar graphs or a 3-connected planar graph ...

We discuss general techniques, centered around the “Layerwise Separation Property” (LSP) of a planar graph problem, that allow to develop algorithms with running time c √ k|G|, given an instance G of a problem on planar graphs with parameter k. Problems having LSP include planar vertex cover, planar independent set, and planar dominating set. Extensions of our speed-up technique to basically al...

We show that the problems of deciding whether an arbitrary graph has a k-regular subgraph, for some given k 3; of deciding whether a planar graph has a quartic subgraph; and of deciding whether a planar graph has a quintic subgraph, are complete for NP via logspace reductions. There are no regular planar graphs of degree greater than 5.

MacGillivray and Seyffarth (J. Graph Theory 22 (1996), 213–229) proved that planar graphs of diameter two have domination number at most three and planar graphs of diameter three have domination number at most ten. They also give examples of planar graphs of diameter four having arbitrarily large domination numbers. In this paper we improve on their results. We prove that there is in fact a uni...

We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph problem—to extend a straight-line planar drawing of a subgraph to a planar drawing of the whole graph—and the simultaneous planarity problem—to find planar drawings of two graphs that coincide on shared vertices and edges. In both cases we show that if the required pl...

A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every edge is crossed by at most one other edge. We show that it is NP-hard to decide whether a given near-planar graph is 1-planar. The main idea in both reductions ...

A plane graph is a planar graph with a fixed planar embedding in the plane. In a boxrectangular drawing of a plane graph, every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal line segment or a vertical line segment, and the contour of each face is drawn as a rectangle. A planar graph is said to have a box-rectangular drawing if at least one of its plane ...