Visibility Representations of Four-Connected Plane Graphs with Near Optimal Heights

A visibility representation of a graph G is to represent the nodes of G with non-overlapping horizontal line segments such that the line segments representing any two distinct adjacent nodes are vertically visible to each other. If G is a plane graph, i.e., a planar graph equipped with a planar embedding, a visibility representation of G has the additional requirement of reflecting the given planar embedding of G. For the case that G is an n-node four-connected plane graph, we give an O(n)-time algorithm to produce a visibility representation of G with height at most n/2+ O(sqrt(n)). To ensure that the first-order term of the upper bound is optimal, we also show an n-node four-connected plane graph G, for infinite number of n, whose visibility representations require heights at least n/2.Acknowledgements.....ihinese Abstract.....iinglish Abstract.....iiiontents.....ivist of Figures.....viist of Tables.....vii Introduction.....1 Preliminaries.....5.1 Ordering and st-ordering.....5.2 Four-canonical ordering.....6.3 Consistent ordering of ladder graph.....7 Our Algorithm.....9.1 Proving Theorem 1.1.....11 A lower bound on the required height.....15ibliography.....1