Series-parallel planar ordered sets have pagenumber two [chapter]

Mohammad Alzohairi, Ivan Rival
1997 Lecture Notes in Computer Science  
The pagenumber of a series-parallel planar P is at most two. We present an O(n 3) algorithm to construct a two-page embedding in the case that it is a lattice. One consequence of independent interest, is a characterization of series-parallel planar ordered sets. I n t r o d u c t i o n A book embedding of a graph G consists of an embedding of its nodes along the spine of a book (i.e., a linear ordering of the nodes), and an embedding of its edges on pages so that edges embedded on the same page
more » ... do not intersect. In a book embedding for an ordered set P the vertices of P on the spine form a linear extension (a total order L = {xl < x2 < ' . . < x~} of the elements of P is a linear extension if x < y in L whenever x < y in P). We say a covers b (or b covered by a) in the ordered set P, and write a ~ b (or b -< a), if whenever a > c > b then c = b. Also, we say a is an upper cover of b, or b is a lower cover of a, or (a, b) is an edge in P. We say a is a minimal (respectively, maxima 0 element of P if a has no lower covers (respectively, a has no upper covers). We denote the set of all minimals (respectively, maximals) of P, rain(P) (respectively, max(P)). The covering graph of P, coy(P), is the graph whose vertices are the elements of P, and the pair {a, b} forms an edge in coy(P) if a ~ b or a -~ b. It is possible to orient cov(P) in such a way the y-coordinate of a is less than the y-coordinate of b if a -~ b and the edge (a, b)
doi:10.1007/3-540-62495-3_34 fatcat:f6ml5fpxireszje4wl7nivtbom