LetGbe a graph. Abook embeddingofGconsists of a total order<σofV(G)and a partition ofE(G)into disjoint sets, calledpages, such that there are no edges(v, w)
1We assume that the edges added tosub(G)to obtainHam(G)all belong to the Hamiltonian cycleCofHam(G). This assumption is not restrictive because a dummy edge that were not in Ccould be removed without altering the Hamiltonicity and the planarity ofHam(G).
and(x, y)in a single page withv <σ x <σw <σy. Atopological book embeddingof Gis a book embedding of a subdivision ofG. A (topological) book embedding withh pages is also called ah-page (topological) book embedding. As observed in [15], every planar graph admits a2-page topological book embedding.
A2-page (topological) book embedding of a planar graphGcan also be regarded as a planar drawing of (a subdivision of)Gsuch that all vertices lie along a horizontal straight line, calledspine, ordered according to<σand each edge is completely drawn in one of the two half-planes defined by the spine. According to this equivalent defi- nition of (topological) book embedding, a division vertex ofGin a topological book embedding ofGis also called aspine crossing.
Starting from a2-page topological book embedding ofGit is immediate to define a Hamiltonian augmentation ofG. Namely, let sub(G) be the subdivision ofGthat defines the topological book embedding and letv1, v2, . . . , vn′be the vertices ofsub(G) in the order they appear along the spine. For every pair of non-adjacent verticesvi, vi+1 we add a dummy edge(vi, vi+1). We also add edge(v0, vn′)if this edge does not exist inG. The cycleC = v0, v1, . . . , vn′is a Hamiltonian cycle of the graph aug(sub(G))consisting ofsub(G)plus the dummy edges, and thereforeaug(sub(G)) is a Hamiltonian augmentation ofG. Also the number of division vertices inHam(G) is equal to the number of spine crossings in the2-page topological book embedding ofG.
Di Giacomo et al. [11] introduce and study a special type of topological book em- bedding. A monotone topological book embedding of a planar graphG is a 2-page topological book embedding such that:
1. each edgee= (u, v)ofGhas at most one spine crossingd;
2. dis encountered betweenuandvwhen walking along the spine;
3. edge(u, d)is below the spine, while edge(d, v)is above the spine.
Starting from a monotone topological book embeddingγofGwe obtain a Hamiltonian augmentation ofGwith at most one division vertex per edge and such that all division vertices are flat. Consider the Hamiltonian pathP = C \(v0, vn′). Each edgeeofG has at most one division vertexdinHam(G)becauseehas at most one spine crossing inγ(Property 1 of the definition of monotone topological book embedding). The two edges ofPincident todare not consecutive in the circular order aroundddefined by the planar embedding ofHam(G)by Property 3 of the definition of monotone topological book embedding. Finally,dis encountered afteruand beforevwhen walking alongP by Property 2 of the definition of monotone topological book embedding.
Theorem 6. [11]Every planar graph withnvertices admits a monotone topological book embedding that can be computed inO(n)time.
Corollary 1. [11]Every planar graph withnvertices admits a Hamiltonian augmen- tation with at most1division vertices per edge such that all division vertices are flat.
This Hamiltonian augmentation can be computed inO(n)time.
By Corollary 1 and Theorem 3 we can answer the first question at the end of Section 2.2.
Theorem 7. LetGbe a planar graph withnvertices and letSbe any set ofndistinct points in the plane. There exists anO(nlogn)time algorithm that computes a point-set embeddingΓ ofGonSwithout mapping and such that the curve complexity is at most 2. Also, the area of the drawing isO(W3), whereW is the length of the side of the smallest axis parallel square containingS.
4 Point-Set Embeddings with Curve Complexity 1
Everett et al. [17] describe set of points that make it possible to point-set embed every planar graph with curve complexity at most one. More precisely, a setSofmpoints is h-bend universalfor a family of planar graphs withnvertices (n ≤m) if each graph in the family admits a point-set embedding on a subset ofSthat has curve complexity at mosth.
Gritzman, Mohar, Pach and Pollack [20] proved that every set ofndistinct points in general position in the plane is0-bend universal for the class of outerplanar graphs withnvertices. De Fraysseix, Pach, and Pollack [8] and independently Schnyder [33]
proved that a grid withO(n2)points is0-bend universal for all planar graphs withn vertices. De Fraysseix et al. [8] also showed that a0-bend universal set of points for all planar graphs havingnvertices cannot haven+o(√n)points. This last lower bound was improved by Chrobak and Karloff [7] and later by Kurowski [29] who showed that linearly many extra points are necessary for a0-bend universal set of points for all planar graphs havingnvertices. On the other hand, if two bends along each edge are allowed, a tight bound on the size of the point-set is implied by Theorem 5. Finally, if one bend along each edge is allowed, Di Giacomo et al. [11] proved a related result that every planar graph can be drawn with its vertices on any given convex curve; however, the positions of the points on the curve depend on the planar graph.
Everett et al. [17] prove the following theorem.
Theorem 8. [17]LetFnbe the family of all planar graphs withnvertices. There exists a set ofndistinct points in the plane in general position that is1-bend universal for Fn.
The proof of Theorem 8 is constructive. A setS ofnpoints is defined and a point-set embedding of a planar graphGon this set of points is constructed by exploiting the Hamiltonian augmentation technique of Corollary 1. Namely, the points are chosen to be in convex position and the Hamiltonian cycleCofHam(G)is drawn as the convex hullCH ofS suitably enriched with extra points that represent the division vertices.
The edges ofHam(G)that are not inC are either insideC or outside it. Those inside are drawn as chords insideCH, the others are drawn with one bend outsideCH. The choice of points and the flatness of the division vertices guarantee that no additional bend is required when the division vertices are removed.
5 Colored Hamiltonicity and Colored Point-Set Embeddability
In this section we present an application of Hamiltonian augmentation techniques to a problem that generalizes and encompasses those described in Sections 2.1 and 2.2.
Namely, we revisit both the point-set embeddability problem with mapping and the one without mapping in the framework of colored point-set embeddability. This new framework is studied by investigating a novel notion of Hamiltonicity, calledcolored Hamiltonicity, and the corresponding colored Hamiltonian augmentation problem. Col- ored Hamiltonicity and colored point-set embeddings have been first introduced by Di Giacomo et al. [13] and further studied in [3,10,12].