Center and Diameter Problems in Plane Triangulations and Quadrangulations

1. Center and Diameter Problems in Plane Triangulations and Quadrangulations Victor D. Chepoi Universite Aix-Marseille II, France Feodor F. Dragan Kent State University, Ohio Yann Vaxes Universite Aix-Marseille II, France `

2. The Center and Diameter Problems • G = (V,E)is a connected, finite, and undirected graph • The length of a path from a vertex v to a vertex u is the number of edges in the path • The distanced(u,v) is the length of a shortest (u,v)-path • The eccentricitye(v) of a vertex v is the maximum distance from v to a vertex in G • The radiusr(G) is the minimum eccentricity of a vertex in G and the diameterd(G) is the maximum eccentricity • The center C(G) of G is the subgraph induced by the set of all central vertices, i.e., vertices whose eccentricities are equal to r(G) • The diameter problem:find d(G) and x,y such that d(x,y)=d(G) • The center problem:find a central vertex v of G or whole C(G)

3. Graphs Considered • Trigraphs:plane triangulations with inner vertices of degree at least six • Squaregraphs: plane quadrangulations with inner vertices of degree at least four • Triangular and Square Systems: the subgraphs of the regular triangular and square grids which are induced by the vertices lying on a simple circuit and inside the region bounded by this circuit. • Benzenoids, alias, Hexagonal Systems: subgraphs of the regular hexagonal grid bounded by a simple circuit • Kinggraphs: the graphs resulting from squaregraphs by transforming each inner face into a 4-clique (includes all subgraphs of the King grid bounded by a simple circuit).

4. Motivation • The diameter and center problems are basic problems in algorithmic graph theory and computational geometry. • They naturally arise in communication and transportation networks,robot-motion planning and also in several other areas. • in distributed systems, centers are ideal locations for placing resources that need to be shared among different processes in a network, • if a graph represents a road network with its vertices representing communities, one may have the problem of locating optimally a hospital, police station, fire station, or any other "emergency“ service facility. • the diameter of a communication network gives a lower bound on the time needed to transmit a message from an arbitrary source node to all other nodes.

5. Motivation (cont.) • The starting point of our investigations of these subclasses of planar graphs was a question: how to find the center (i.e., all central vertices) of a benzenoid system efficiently [A. Balaban; in his H. Skolnik award lecture]. • Benzenoids represent a significant class of chemical graphs and their encoding constitutes an important subject of research in computational chemistry. • One canonical way of such an encoding is to label the carbon atoms level-wise starting from the center. • Trying to find an efficient algorithm for the center problem on benzenoids, we noticed that its solution can be obtained from the solution of the same problem on two triangular systems inferred from the initial benzenoid

6. Motivation (cont.) • Squaregraphscan be used to model a road network (streets) in a city. • Squaregraphs and Kinggraphs contain as particular cases two important classes of discrete metric spaces, extensively studied in digital geometry: • simply connected sets of lattice points in the plane under the graph structure defined by 4-neighbor adjacency (city block distance) or 8-neighbor adjacency (chessboard distance) • Such sets of lattice points ("pixels") arise when planar regions are digitized; they can be regarded as discrete approximations of these regions.

7. Our Results • We designed a general approach for computing the centers and diameters which can be applied not only to triangular systems but also to all trigraphs, squaregraphs and kinggraphs. • It gives linear time algorithms for computing the centers and diameters in all those graph classes as well as in all benzenoids. • Additionally, we characterized centers of trigraphs and kinggraphs (answering a question posed by Khuller et al. [KhRoWu’00]). • The centers of squaregraphs were characterized by Postaru [Po’84] and, independently, by Khuller et al. in [KhRoWu’00] for the particular case of the square systems. • Some properties of centers of square systems have been given by Metivier and Saheb in [MeSa’96]. • No linear time algorithms were known for computing the centers and diameters of those graphs. . . .

8. Method: The Diameter Problem • For any vertex v of G, the set F(v)={ u V: d(v,u)=e(v) } of furthest neighbors belong to . ([Lyndon’67] and [Baues and Peyerimhoff’01]) • To find we use row-wise maxima search of Aggarwal et al. [AKMSW’87] in totally monotone matrix. • matrix Dis totally monotone if D(i,k)<D(i,l)implies D(j,k)<D(j,l)for any • The matrix is defined implicitly -- an entry is evaluated only when needed by the algorithm. If evaluating an entry takes O(f(n,m)) time, then the complexity of the algorithm is O((n+m)f(n,m)). is totally monotone matrix

9. d(p,q) in constant time after linear time preprocessing • Get metric interval I(v,w)={ x V: d(v,x)+d(x,w)=d(v,w)}.It is a convex set and induces a triangular system. • Get distances and projections of vertices from P and Q to I(v,w). • Embed I(v,w) isometrically into the Cartesian product of three trees in linear time. Then, for any p,q, the distance can be computed in constant time.

10. Diameters of Squaregraphs and Kinggraphs • Squaregraphs: in a similar way as for trigraphs, even easier. • Kinggraphs: we reduce the problem to the squaregraphs K Q(K) • Theorem 1: Diameters of trigraphs, squaregraphs, kinggraphs and benzenoids can be computed in linear time. • The idea to use matrix-searching to compute the diameter of a simple polygon was employed by Hershberger and Suri in [HeSu’97].

11. Method: The Center Problem • Having a diametral pair, a simple region containing at least one central vertex will be located and preprocessed in such a way that all vertices of minimum eccentricity in this region can be found in linear time. • Then, using the established structure of the center, the remaining part of the center can build up. Get histogram H(c) Get convex cut c of trigraph T A histogramH(c) of a convex cut c is the union of all metric triangles having one side on c. It is an isometric subgraph of T (and of triangular grid).

12. Method: The Center Problem • Get distances and projections of vertices from to H(c) • harder, since H(c) is not necessarily convex; it is convex if T does not contain inner vertices of degree 7. • the case with inner vertices of degree 7 is handled separately. • Embed H(c) into triangular grid and consider quadrangles. • Find vertices of least eccentricity in each quadrangle (reduced to analyzing a system of at most 6 inequalities with two variables and one unknown parameter). • Thus, can be found in linear time.

13. Centers of Trigraphs • Structure of the center: The center of a trigraph T is a 3-sun, a convex path, or a convex strip. • Using the established structure of the center, one can get the entire center C(T) from the set in linear time. • Here is the idea how we do this for a triangular systemT • Let • Consider disk B(w,2) (3-sun case) • Consider or intersections of C(T) with 9 convex cuts.

14. Centers of Squaregraphs, Kinggraphs and Benzenoids • Squaregraphs: in a similar way as for trigraphs. Even easier since: • Kinggraphs and Benzenoids: The centers of hexagonal systems and kinggraphs are obtained by simply employing their relation to triangular systems and squaregraphs, respectively. • The center of a kinggraph K is an isometric path or an isometric chain of . • Theorem 2: Centers of trigraphs, squaregraphs, kinggraphs and benzenoids can be found in linear time.

15. Concluding Remarks and Open Problems • We presented a general approach for computing in linear time the centers and diameters of trigraphs, benzenoid systems, squaregraphs and kinggraphs. • We characterized centers of trigraphs and kinggraphs (answering a question posed by Khuller et al. [KhRoWu’00]). • Fewinteresting open problemsremain: • Characterize the centers of benzenoid systems. • To which other face regular planar graphs can this method be applied? • Can it be extended to 3-dimensional variants of squaregraphs, kinggraphs and trigraphs? • Can the p-center problem (p=2,3,…) be solved efficiently on those classes of graphs? . . .