Paul C. Kainen. Thickness and Coarseness of Graphs. Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 39:88--95, 1973.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Figure 12: Three planar subgraphs G 1 , G 2 , and G 3 of K 9 whose union is K 9 . labeled i in the second subgraph. So we do not only have two subgraphs whose union is K 3,3 , but we have two embeddings of two planar graphs so that the union of the embeddings yields a drawing of K 3,3 . Kainen [Kai73] showed that this observation can be generalized: Theorem 36 [Kai73] Given a graph G with thickness #(G) there exists a drawing of G, and there exist subgraphs G 1 , G #(G) whose union is G, such that the drawing of G restricted to G i is a planar embedding of G i , for 1 # i # #(G) ....

....union is K 9 . labeled i in the second subgraph. So we do not only have two subgraphs whose union is K 3,3 , but we have two embeddings of two planar graphs so that the union of the embeddings yields a drawing of K 3,3 . Kainen [Kai73] showed that this observation can be generalized: Theorem 36 [Kai73] Given a graph G with thickness #(G) there exists a drawing of G, and there exist subgraphs G 1 , G #(G) whose union is G, such that the drawing of G restricted to G i is a planar embedding of G i , for 1 # i # #(G) Note that the three subgraphs of K 9 in Figure 12 are drawn in a way ....

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Paul C. Kainen. Thickness and Coarseness of Graphs. Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 39:88--95, 1973.

....smallest value of k such that we can assign planar point locations to the vertices of G, represent each edge of G as a line segment, and assign each edge to one of k layers so that no two edges on the same layer cross. This corresponds to the notion of real linear thickness introduced by Kainen [15]. Graphs with geometric thickness 2 (called doubly linear graphs) have been studied by Hutchinson et al. 13] where the connection with certain types of visibility graphs was explored. A notion related to geometrical thickness is that of (graph theoretical) thickness of a graph, #(G) which has ....

....graph theoretical thickness imposes no consistency requirement between layers. Alternatively, the graph theoretical thickness can be defined as the minimum number of planar layers required to embed a graph such that the vertex placements agree on all layers but the edges can be arbitrary curves [15]. The equivalence of the two definitions follows from the observation that, given any planar embedding of a graph, the vertex locations can be reassigned arbitrarily in the plane without altering the topology of the planar embedding provided we are allowed to bend the edges at will [15] This ....

[Article contains additional citation context not shown here]

P. C. Kainen. Thickness and coarseness of graphs. Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 39:88--95, 1973.

....smallest value of k such that we can assign planar point locations to the vertices of G, represent each edge of G as a line segment, and assign each edge to one of k layers so that no two edges on the same layer cross. This corresponds to the notion of real linear thickness introduced by Kainen [10]. A related notion is that of (graph theoretical) thickness of a graph, G) which has been studied extensively [1, 5, 6, 7, 9, 11] and has been defined as the Supported by NSF Grants CDA 9617349 and CCR 9703572. Supported by NSF Grant CCR 9258355 and matching funds from Xerox Corp. ....

....graph theoretical thickness imposes no consistency requirement between layers. Alternatively, the graph theoretical thickness can be defined as the minimum number of planar layers required to embed a graph such that the vertex placements agree on all layers but the edges can be arbitrary curves [10]. The equivalence of the two definitions follows from the observation that, given any planar embedding of a graph, the vertex locations can be reassigned arbitrarily in the plane without altering the topology of the planar embedding provided we are allowed to bend the edges at will [10] This ....

[Article contains additional citation context not shown here]

P. C. Kainen. Thickness and coarseness of graphs. Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 39:88--95, 1973.

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