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
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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(d1, d2, d3,
  ,dk)-Coloring or Distance-k-Coloring of a graph.

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L(d1, d2 ,d3 ,,dk)- coloring

• L(d1, d2 ,d3 ,,dk)- coloring of a graph G(V,
  E), where dis are positive integers, is an
  assignment function ? V ? N?0 such that ?(u)
  - ?(v) ? di when the distance between u and v in
  G is equal to i (i?1,2,,k). The aim is to
  minimize ? such that G admits a L(d1, d2 ,d3
  ,,dk)- coloring with frequencies/colors between
  0 and ?.

Examples of L(2,1) coloring. Each color is
associated with a unique integer number
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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 (k1) Reuse coloring
• The relationship between L(d1, d2 ,d3 ,,dk)-
  coloring and Distance-k-coloring is that in
  Distance-k-coloring di is set to 1, for i1, 2,
  , k.

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New r-coloring and r-coloring

• Let r V ? NU0 be a radius-function defined
  on V.
• We define r-coloring of G as an assignment ?
  V?0,1,2, of colors to vertices such that ?(u)
  ?(v) implies dG(u,v)gtr(v)r(u), and r-coloring
  of G as an assignment ? V?0,1,2, of colors to
  vertices such that ?(u) ?(v) implies
  dG(u,v)gtr(v)r(u)1.
• This is a new formulation which generalizes the
  Distance-k-Coloring, approximates L(d1, d2 ,d3
  ,,dk)-coloring, and is suitable for
  heterogeneous multihop radio networks.
• Let t max1ikdi . From a valid
  Distance-k-Coloring, one can get a
• L(d1, d2 ,d3 ,,dk)-coloring by multiplying
  each integer/color by t.

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Old Powers of Graphs

• Given an 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
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New Generalized Powers of Graphs

• Given an unweighted graph G(V, E) and a radius
  function r V?NU0
• (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
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New Generalized Powers of Graphs

• Given an unweighted graph G(V, E) and a radius
  function r V?NU0
• (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
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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 G 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
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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
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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
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Our Results

• Theorem 2. For a graph G, is
  weakly chordal if and only if G2 is weakly
  chordal.

0
1
1
0
0
1
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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.

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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

0
0
0
3
Unit intervals
Unit interval graph with r values
L graph (not unit interval graph)
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Complexity results for the r-Coloring and
r-Coloring problems on several graph families
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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 G graphs.
• This gives also approximation algorithms for the
  L(d1 , d2 ,d3 ,,dk)-coloring problem on those
  families of graphs.

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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?

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