1. Generalized Powers of Graphs and their Algorithmic Use A. Brandstädt, F.F. Dragan, Y. Xiang, and C. Yan University of Rostock, Germany Kent State University, Ohio, USA

2. Frequency Assignment Problem The Frequency Assignment Problem (FAP) in multi-hop radio networks is the problem of assigning frequencies to transmitters exploiting frequency reuse while keeping signal interference to acceptable levels. FAP can be viewed as a variant of the graph coloring problem. Frequency Assignment Problem in wireless networks is usually modeled as L(δ 1, δ 2, δ 3, …,δ k )-Coloring or Distance-k-Coloring of a graph.

3. L(δ 1, δ 2,δ 3,…,δ k )- coloring L(δ 1, δ 2,δ 3,…,δ k )- coloring of a graph G=(V, E), where δ i s are positive integers, is an assignment function Ф: V  N ∪ {0} such that |Ф(u) - Ф(v)|  δ i when the distance between u and v in G is equal to i (i ∈ {1,2,…,k}). The aim is to minimize λ such that G admits a L(δ 1, δ 2,δ 3,…,δ k )- coloring with frequencies/colors between 0 and λ. Examples of L(2,1) coloring. Each color is associated with a unique integer number

4. Distance-k-Coloring Distance-k-Coloring is defined as coloring of G k, the kth power of G, with minimum number of colors. Two vertices v and u are adjacent in G k if and only if their distance in G is at most k. –Distance (k+1) Reuse coloring The relationship between L(δ 1, δ 2,δ 3,…,δ k )- coloring and Distance-k-coloring is that in Distance-k-coloring δ i is set to 1, for i=1, 2, …, k.

5. New r-coloring and r + -coloring ∙ Let r : V → NU{0} be a radius-function defined on V. ∙ We define r-coloring of G as an assignment Ф: V  {0,1,2,…} of colors to vertices such that Ф(u) = Ф(v) implies d G (u,v)>r(v)+r(u), and r + -coloring of G as an assignment Ф: V  {0,1,2,…} of colors to vertices such that Ф(u) = Ф(v) implies d G (u,v)>r(v)+r(u)+1. ∙ This is a new formulation which generalizes the Distance-k-Coloring, approximates L(δ 1, δ 2,δ 3,…,δ k )-coloring, and is suitable for heterogeneous multihop radio networks.  Let t = max 1≤i≤k {δ i }. From a valid Distance-k-Coloring, one can get a L(δ 1, δ 2,δ 3,…,δ k )-coloring by multiplying each integer/color by t.

6. Old Powers of Graphs Given an unweighted graph G=(V, E) and an integer k G k =(V, E’) is kth power of G, if for any two vertices u, v in G, {u, v} is in E’ if and only if d G (u, v)≤k Original graph G G2G2

7. New Generalized Powers of Graphs Given an unweighted graph G=(V, E) and a radius function r : V→NU{0} =(V, E’) (generalized powers of G): for any two vertices u, v in G, {u, v} is in E’ if and only if d G (u, v)≤r(u)+r(v)  the intersection graph of the family of disks is defined as the intersection graph of the family of disks Original graph G 1 0 1 1 11

8. New Generalized Powers of Graphs Given an unweighted graph G=(V, E) and a radius function r : V→NU{0} =(V, E’) (generalized powers of G): for any two vertices u, v in G, {u, v} is in E’ if and only if d G (u, v)≤r(u)+r(v)+1  the visibility graph of the family of disks is defined as the visibility graph of the family of disks Original graph G 1 0 1 1 11 )),((rGD 

9. Use of Generalized Powers of Graphs Generalization of the old notion of the kth power of a graph To solve the r-coloring or r + -coloring problem on graph G, we can first create L graph or Γ graph of the original graph and then apply some known coloring algorithms on them. Can be used to assign frequencies in heterogeneous multi-hop networks. Original graph G 1 0 1 1 11

10. c-Chordal Graphs A graph G is c-chordal if the length of its largest induced cycle is at most c  A 3-chordal graph is also called a chordal graph 3-chordal graph4-chordal graph

11. Our Results Theorem 1. For a graph G, is weakly chordal if and only if G is weakly chordal (A graph is weakly chordal if and only if G and its complement are 4-chordal) 1 0 0 0

12. Our Results Theorem 2. For a graph G, is weakly chordal if and only if G 2 is weakly chordal. 1 0 0 1 01

13. Our Results Theorem 3. Let G = (V, E) be an AT-free graph and r : V → N be a radius-function defined on V. Then, both and are co-comparability graphs. Theorem 4. Let G= (V, E) be a co-comparability graph. Then, for any radius-function r: V  N, is a co-comparability graph, and for any radius-function r: V  N ∪ {0}, is a co-comparability graph. Theorem 5. Let G=(V, E) be an interval graph. Then, for any radius-function r: V  N, is an interval graph, and for any radius-function r: V  N ∪ {0}, is an interval graph.

14. Results on ordinary powers cannot always be extended to generalized powers It is well-known that all powers of unit interval graphs are unit interval graphs The L graphs of unit interval graphs are no longer unit interval graphs 0003 Unit intervals Unit interval graph with r values L graph (not unit interval graph)

15. Complexity results for the r-Coloring and r + -Coloring problems on several graph families

16. Conclusion r-Coloring ( r + -Coloring ) is NP-complete in general. But, as we show, for many graph families, the problem can be solved in polynomial time, by applying known coloring algorithms to L graphs or Γ graphs. This gives also approximation algorithms for the L(δ 1, δ 2,δ 3,…,δ k )-coloring problem on those families of graphs.

17. In journal version We show also that for any circular-arc graph G and any radius-function r: V  N, both graphs and are circular-arc, too. We discuss other applications of the generalized powers of graphs (e.g. to r-packing, q-dispersion, k- domination, p-centers, r-clustering, etc.) What is the complexity of r-coloring for circular-arc graphs, other graphs? Open