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Generalized Powers of Graphs and their Algorithmic Use

Generalized Powers of Graphs and their Algorithmic Use. A. Brandst ädt , F.F. Dragan , Y. Xiang, and C. Yan. University of Rostock, Germany Kent State University, Ohio, USA. Frequency Assignment Problem.

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Generalized Powers of Graphs and their Algorithmic Use

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  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

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