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The book thickness of a graph is the smallest possible number of half-planes for any book embedding of the graph. Book thickness is also called pagenumber, stacknumber or fixed outerthickness. Book embeddings have also been used to define several other graph invariants including the pagewidth and book crossing number.
[1] [2] The 7-page book graph of this type provides an example of a graph with no harmonious labeling. [2] A second type, which might be called a triangular book, is the complete tripartite graph K 1,1,p. It is a graph consisting of triangles sharing a common edge. [3] A book of this type is a split graph.
2. Another type of graph, also called a book, or a quadrilateral book, is a collection of 4-cycles joined at a shared edge; the Cartesian product of a star with an edge. 3. A book embedding is an embedding of a graph onto a topological book, a space formed by joining a collection of half-planes along a shared line. Usually, the vertices of the ...
In 2012, the head of AP Grading, Trevor Packer, stated that the reason for the low percentages of 5s is that "AP World History is a college-level course, & many sophomores aren't yet writing at that level." 10.44 percent of all seniors who took the exam in 2012 received a 5, while just 6.62 percent of sophomores received a 5.
Twin-width is defined for finite simple undirected graphs. These have a finite set of vertices, and a set of edges that are unordered pairs of vertices. The open neighborhood of any vertex is the set of other vertices that it is paired with in edges of the graph; the closed neighborhood is formed from the open neighborhood by including the vertex itself.
An embedded graph uniquely defines cyclic orders of edges incident to the same vertex. The set of all these cyclic orders is called a rotation system.Embeddings with the same rotation system are considered to be equivalent and the corresponding equivalence class of embeddings is called combinatorial embedding (as opposed to the term topological embedding, which refers to the previous ...
AP journalist Stephen Smith contributed to this report. The Associated Press education team receives support from the Carnegie Corporation of New York. The AP is solely responsible for all content.
According to Jensen & Toft (1995), the problem was first formulated by Nelson in 1950, and first published by Gardner (1960). Hadwiger (1945) had earlier published a related result, showing that any cover of the plane by five congruent closed sets contains a unit distance in one of the sets, and he also mentioned the problem in a later paper (Hadwiger 1961).