Geometric Graphs and Quasi-Planar Graphs Dafna Tanenzapf

Definition – Geometric Graph A geometric graph is a graph drawn in the plane by straight-line segments . There are no three points in which are collinear. The edges of can be possibly crossing.

Forbidden Geometric Graphs Given a class H of forbidden geometric graphs determine (or estimate) the maximum number of edges that a geometric graph with n vertices can have without containing a subgraph belonging to H.

Definition Let be the class of all geometric graphs with vertices, consisting of pairwise disjoint edges ( ). Example (k=2):

Theorem Let be the maximum number of edges that a geometric graph with n vertices can have without containing two disjoint edges (straight-line thrackle). Then for every :

Theorem (Goddard and others) Let be the maximum number of edges that a geometric graph with n vertices can have without containing three disjoint edges. Then:

Definitions Edge xy is Leftmost at x if we can rotate it by (180 degrees) counter-clockwise around x without crossing any other edge of x. A vertex x is called pointed if it has an angle between two consecutive edges that is bigger than . y z x x

Proof Let G be a geometric graph with n vertices and at least 3n+1 edges. We will show that there exist three edges which are disjoint.

Proof - Continue For each pointed vertex at G, delete the leftmost edge. We will denote the new subgraph as G1. For each vertex at G1 delete an edge if there are no two edges from its right (begin from the leftmost edge of every vertex).

Proof - continue Examples: |E|=1 |E|=2 |E|=3 |E|≥4 Number of deleted edges: 1 2 3 3 z z u v y y u y z y x x x x

Proof - continue For every vertex we deleted at most 3 edges. The remaining graph has at least one edge x0y0 (3n+1-3n=1). In G1 there were two edges to the right of x0y0: x0y1 and x0y2. y0x1 and y0x2.

Proof - continue In G there was the edge x2y to the left of x2y0. In G there was the edge y2x to the left of y2x0. WLOG we can assume that the intersection of x0y2 and y0x2 is on the same side of y2 (if we split the plane into two parts by x0y0). The edges y0x2, x0y1, y2x don’t intersect.

Conclusions The upper bound is: The lower bound is:

What’s next? Dilworth Theorem Pach and Torocsik Theorem

Definition – Partially ordered sets A partially ordered set is a pair . X is a set. is a reflexive, antisymmetric and transitive binary relation on X. are comparable if or . If any two elements of a subset are comparable, then C is a chain. If any two elements of a subset are incomparable, then C is an antichain.

Theorem (Dilworth) Let be a finite partially ordered set. If the maximum length of a chain is k, then X can be partitioned into k antichains. If the maximum length of an antichain is k, then X can be partitioned into k chains.

Explanation (Dilworth Theorem) If the maximum length of a chain is k, then X can be partitioned into k antichains. For any x X, define the rank of x as the size of the longest chain whose maximal element is x. 1≤rank(x)≤k The set of all elements of the same rank is an antichain. If the maximum length of an antichain is k, then X can be partitioned into k chains. It can be shown by induction on k.

Definitions Let uv and u’v’ be two edges. For any vertex v let x(v) be the x-coordinate and let y(v) be the y-coordinate. Suppose that x(u)<x(v) and x(u’)<x(v’). uv precedes u’v’ (uv<<u’v’) if x(u) x(u’) and x(v) x(v’). The edge uv lies below u’v’ if there is no vertical line l that intersects both uv and u’v’ such that: y(l∩uv) y(l∩u’v’). u’ v’ v u u’ v’ u v l

Theorem (Pach and Torocsik) Let denote the maximum number of edges that a geometric graph with n vertices can have without containing k+1 pairwise disjoint edges. Then for every k,n 1:

Proof Let G be a geometric graph with n vertices, containing no k+1 pairwise disjoint edges. WLOG, no two vertices of G have the same x-coordinate. Let uv and u’v’ be two disjoint edges of G such that uv lies below u’v’.

Proof - continue We define four binary relations on E(G): uv <1 u’v’ if uv<<u’v’. uv <2 u’v’ if u’v’<<uv. uv <3 u’v’ if [x(u),x(v)] [x(u’),x(v’)]. uv <4 u’v’ if [x(u’),x(v’)] [x(u),x(v)].

Proof - continue

Proof - continue We can conclude from the definitions that: is a partially ordered set . Any pair of disjoint edges is comparable by at least one of the relations .

Proof - continue cannot contain a chain of length k+1. Otherwise, G has k+1 pairwise disjoint edges. According to the previous theorem, for any i, E(G) can be partitioned into at most k anti-chains (classes), so that no two edges belonging to the same class are comparable by . The edges can be partitioned into classes Ej , such that no two elements of Ej are comparable by any relation.

Proof - continue From the second conclusion, any Ej does not contain two disjoint edges. We saw earlier that . Then: To sum up:

Definition – Quasi Planar Graphs A graph is called quasi planar if it can be drawn in the plane so that no three of its edges are pairwise crossing.

Motivation We want to find an upper bound to the number of edges of quasi-planar graph. Pach had shown that a quasi-planar graph with n vertices has edges. For the general case, k-quasi-planar graph (a graph with no k pairwise crossing edges), the upper bound is: We will prove a Theorem. Its conclusion will be that the upper bound is: (can be shown in induction).

Theorem (Agarwal, Aronov, Pach, Pollack and Sharir) If G(V,E) is a quasi-planar graph (undirected, without loops or parallel edges), then |E|=O(|V|). We will prove the theorem for the case that G has a straight-line drawing in the plane with no three pairwise crossing edges.

Definition The arrangement A(E) of E(G) is a complex set consisted of: Nodes: N={V(G) X(G)|X(G)=crossing points}. Segments: S={E’(G)|E’(G)=the edges between the vertices of N}. Faces F={the faces of the graph G’=(N,S)}.

Definition The complexity |f| of f F is the number of segments in S on the boundary f of f. If a segment is in the interior of f, then it contributes 2 to |f|.

Lemma2 Let G=(V,E) be a quasi-planar graph drawn in the plane. The complexity of all f of A(E) such that: f is a non-quadrilateral face. f is quadrilateral face incident to at least on vertex of G. is O(|V|+|E|).

Definition A graph is called overlap graph if its vertices can be represented by intervals on a line so that two vertices are adjacent if and only if the corresponding intervals overlap but neither of them contains the other.

Proof (Theorem) Let G be a quasi-planar graph drawn in the plane with n vertices. WLOG G is a connected graph. Let G0=(V,E0) be a spanning tree of G, |E0|=n-1 E*=E\E0. G: G0:

Proof - continue Each face of A(E0) is simply connected. By Lemma 2, the complexity of non-quadrilateral and quadrilateral faces of A(E0) incident to a point of V is O(n). We call the remaining faces of A(E0) crossing quadrilateral.

Proof - continue For each edge e E*, let Ω(e) denote the set of segments of A(E0 {e}) that are contained in e. Every s Ω(e) is fully contained in some face f A(E0) and its two endpoints lie on f.

Proof - continue For each f A(E0) let X(f) denote the set of all segments in that are contained in f. Any two segments in X(f) cross each other if and only if their endpoints alternate along f. We “cut” the face in some vertex and get an open interval. Two elements of X(f) cross each other if and only if the corresponding intervals overlap.

Proof - continue This defines a triangle-free overlap graph on the vertices X(f). By Gyarfas and Kostochka, every triangle-free overlap graph can be colored by 5 colors. Therefore, the segments of X(f) can be colored by at most 5 colors, so that no two segments with the same color cross each other.

Proof - continue For each f A(E0) let H(f) denote the quasi- planar graph whose edges are X(f). A monochromatic graph is a graph which all its edges are colored in one color. Let f A(E0) be a face that is not crossing quadrilateral. Let H1(f),…,H5(f) be the monochromatic subgraphs of H.

Proof - continue We fix one of Hi, assume WLOG H1, and we reinterpret it to a new graph: every edge on the boundary of the face and vertices of V on the boundary will be a vertex, and all the interior segments will be the edges in the new graph. The resulting graph H1* is planar. b b a c c a d p e p e d

Proof - continue Face of H1*(f) is a digon if it is bounded by two edges of H1*(f). Edge of H1*(f) is shielded if both of the faces incident to it are digons. The remaining edges of H1*(f) are called exposed. b b a c c a d p e p e d

Proof - continue By Euler’s formula (|E|≤3|V|-6), there are at most O(nf) exposed edges in H1*(f). nf is the number of vertices of H1*(f). |nf|≤2|f|.

Proof - continue We repeat this analysis for every Hi(f) (2≤i≤5). The number of edges e E* containing at least one exposed segment is . By Lemma2, this sum is O(n). We need to bound the number of e E* with no exposed segments (shielded edges).

Lemma 3 There are no shielded edges.

Proof - continue The total number of edges of E* is O(n).