Generalized Powers of Graphs and their Algorithmic Use

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(δ1, δ2, δ3, …,δk)-Coloring or Distance-k-Coloring of a graph.

3. L(δ1, δ2 ,δ3 ,…,δk)- coloring • L(δ1, δ2 ,δ3 ,…,δk)- coloring of a graph G=(V, E), where δis are positive integers, is an assignment function Ф: V  N∪{0} such that |Ф(u) - Ф(v)|  δiwhen 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(δ1, δ2 ,δ3 ,…,δk)- 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 Gk, the kth power of G, with minimum number of colors. Two vertices v and u are adjacent in Gk if and only if their distance in G is at most k. • Distance (k+1) Reuse coloring • The relationship between L(δ1, δ2 ,δ3 ,…,δk)- coloring and Distance-k-coloring is that in Distance-k-coloring δi is set to 1, for i=1, 2, …, k.

5. New r-coloring and r+-coloring • Letr : 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 dG(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 dG(u,v)>r(v)+r(u)+1. • This is a new formulation which generalizes the Distance-k-Coloring, approximates L(δ1, δ2 ,δ3 ,…,δk)-coloring, and is suitable for heterogeneous multihop radio networks. • Let t = max1≤i≤k{δi} . From a valid Distance-k-Coloring, one can get a L(δ1, δ2 ,δ3 ,…,δk)-coloring by multiplying each integer/color by t.

6. OldPowers of Graphs • Givenan unweighted graph G=(V, E) and an integer k • Gk=(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 dG(u, v)≤k G2 Original graph G

7. NewGeneralized Powers of Graphs • Givenan 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 dG(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 1 1 0 1 1 1 Original graph G

8. G ( D ( G , r )) NewGeneralized Powers of Graphs • Givenan 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 dG(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 1 1 0 1 1 1 Original graph G

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. 1 1 0 1 1 1 Original graph G

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 graph 4-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 G2 is weakly chordal. 0 1 1 0 1 0

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 3 0 0 0 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(δ1 , δ2 ,δ3 ,…,δk)-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