1. Fixed-Parameter Algorithms for (k,r)-Center in Planar Graphs and Map Graphs Erik D. Demaine, Fedor V. Fomin, MohammadTaghi Hajiaghayi, and Dimitrios M. Thilikos ACM Transactions on Algorithms, volume 1, number 1, July 2005, pages 33-47 Jeryann Huang

2. (k,r)-center problem Given an unweighted graph G and asks whether G has ≤ k vertices (centers) such that every vertex of G is within distance ≤ r from some center. Results:  For planar graphs, the running time is  For map graphs, the running time is where n is the number of vertices

3. Previous Results Alber et al. [2002]: Kanj and Perkovi´c [2002]: Fomin and Thilikos [2003]: Most of these problems have reductions to the dominating set problem.

4. Minor A graph G has a minor H if H can be formed by removing and contracting edges of G Otherwise, G is H-minor-free For example, planar graphs are both K 3,3 -minor-free and K 5 -minor-free delete contract

5. Map Graphs THEOREM 2.1 [CHEN ET AL. 2002]. A graph G M is a map graph if and only if it is the half-square of some planar bipartite graph H.  Graph H is a witness for G M.  Finding a (k, r )-center in a map graph G M is equivalent to finding in a witness H of G M a set S ⊆ V(G M ) of size k such that every vertex in V(G M ) − S has distance ≤ 2r in H from some vertex of S

6. Branchwidth Branch decomposition: A pair (T, v), where T is a tree with vertices of degree 1 or 3 and v is a bijection from E(G) to the set of leaves of T order function ω : E(T ) → 2 V(G) of a branch decomposition maps every edge e of T to a subset of vertices ω(e) ⊆ V(G) 01 2 3 G e 4 0 1 2 3 4 (T, )

7. Branchwidth (cont.) The width of (T, v) is equal to max e ∈ E(T ) |ω(e)| and the branchwidth of G, bw(G), is the minimum width over all branch decompositions of G. G a b d e f g c b g d c a f e 33 2 3 2 2 2 2 2 2 4 width = 4

8. Combinatorial Bounds LEMMA 3.1. Let p, k, r ≥ 1 be integers and G be a planar graph having a (k, r )-center and with a (p × p)-grid as a minor. Then k ≥ ( p−2r / 2r+1 ) 2. THEOREM 2.2 [ROBERTSON ET AL. 1994] G contains a (p ×p)-grid as a minor where p = (2r + 1)√k + 2r + c/4 for some c, 0 < c <= 4 p p

9. Combinatorial Bounds (cont.) THEOREM 3.2. For any planar graph G having a (k, r )-center, bw(G) ≤ 4(2r+ 1)√k + 8r + 1 THEOREM 3.3. For any map graph G M having a (k, r )-center and its witness H, bw(H) ≤ 4(4r + 3)√k + 16r + 9

10. Algorithms Step1: Check whether the branchwidth of G is at most 4(2r +1)√k+8r +1  If the answer is negative, report no and stop.  Running time: O((|V(G)| + |E(G)|) 2 ) Step2: Compute an optimal branch- decomposition of graph G  Running time: O((|V(G)| + |E(G)|) 4 ) Step 3: Compute, if is exists, a (k,r)-center of G using dynamic-programming algorithm

11. Results Dynamic program are defined by a coloring of the vertices in ω( f ) for every edge f of T. Each vertex will be assigned one of 2r + 1 colors {0, ↑1, ↑2,..., ↑r, ↓1, ↓2,..., ↓r }  0 means that the vertex v is a chosen center.  ↓i means that vertex v has distance exactly i to the closest center c with a neighbor that has distance i-1  ↑i means that vertex v has distance exactly i to the closest center c with no neighbor

12. Results (cont.) For a graph G on m edges and with a given branch decomposition of width ≤ L, and integers k, r, the existence of a (k, r )-center in G can be checked in O((2r + 1) 3/2*L ·m) time and, in case of a positive answer, constructs a (k, r )- center of G in the same time. For planar graph, L= 4(2r+ 1)√k + 8r + 1 Running time = For map graph, L = 4(4r + 3)√k + 16r + 9 Running time =