# Strong Orientations of Planar Graphs with Bounded Stretch Factor

• Evangelos Kranakis
• Oscar Morales Ponce
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6058)

## Abstract

We study the problem of orienting some edges of given planar graph such that the resulting subdigraph is strongly connected and spans all vertices of the graph. We are interested in orientations with minimum number of arcs and such that they produce a digraph with bounded stretch factor. Such orientations have applications into the problem of establishing strongly connected sensor network when sensors are equipped with directional antennae.

We present three constructions for such orientations. Let G = (V, E) be a connected planar graph without cut edges and let Φ(G) be the degree of largest face in G. Our constructions are based on a face coloring, say with λ colors. First construction gives a strong orientation with at most $$\left( 2 - \frac{4 \lambda - 6}{\lambda (\lambda - 1)} \right) |E|$$ arcs and stretch factor at most Φ(G) − 1. The second construction gives a strong orientation with at most |E| arcs and stretch factor at most $$(\Phi (G) - 1)^{\lceil \frac{\lambda + 1}{2} \rceil}$$. The third construction can be applied to planar graphs which are 3-edge connected. It uses a particular 6-face coloring and for any integer k ≥ 1 produces a strong orientation with at most $$\left(1 - \frac{k}{10 (k + 1)}\right) |E|$$ arcs and stretch factor at most Φ2 (G) (Φ(G) − 1)2 k + 4. These are worst-case upper bounds. In fact the stretch factors depend on the faces being traversed by a path.

## Keywords and Phrases

Digraph Directional Antennae Planar Sensors Cut Edges Spanner Stretch Factor Strongly Connected

## Preview

Unable to display preview. Download preview PDF.

## References

1. [1]
Bhattacharya, B., Hu, Y., Kranakis, E., Krizanc, D., Shi, Q.: Sensor network connectivity with multiple directional antennae of a given angular sum. In: IEEE IPDPS, 23rd International Parallel and Distributed Processing Symposium, May 25-29, pp. 344–351 (2009)Google Scholar
2. [2]
Borodin, O.V.: On acyclic colorings of planar graphs. Discrete Math. 25(3), 211–236 (1979)
3. [3]
Caragiannis, I., Kaklamanis, C., Kranakis, E., Krizanc, D., Wiese, A.: Communication in wireless networks with directional antennas. In: SPAA 2008: Proceedings of the twentieth annual symposium on Parallelism in algorithms and architectures, pp. 344–351. ACM Press, New York (2008)
4. [4]
Czyzowicz, J., Dobrev, S., Gonzalez-Aguilar, H., Kralovic, R., Kranakis, E., Opatrny, J., Stacho, L., Urrutia, J.: Local 7-coloring for planar subgraphs of unit disk graphs. In: Agrawal, M., Du, D.-Z., Duan, Z., Li, A. (eds.) TAMC 2008. LNCS, vol. 4978, pp. 170–181. Springer, Heidelberg (2008)
5. [5]
Dobrev, S., Kranakis, E., Krizanc, D., Morales, O., Opatrny, J., Stacho, L.: Strong connectivity in sensor networks with given number of directional antennae of bounded angle (2009) (to appear)Google Scholar
6. [6]
Fukunaga, T.: Graph orientations with set connectivity requirements. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 265–274. Springer, Heidelberg (2009)
7. [7]
Gupta, P., Kumar, P.R.: The capacity of wireless networks. IEEE Transactions on Information Theory 46(2), 388–404 (2000)
8. [8]
Jensen, T., Toft, B.: Graph coloring problems. Wiley-Interscience, New York (1996)Google Scholar
9. [9]
Kranakis, E., Krizanc, D., Williams, E.: Directional versus omnidirectional antennas for energy consumption and k-connectivity of networks of sensors. In: Higashino, T. (ed.) OPODIS 2004. LNCS, vol. 3544, pp. 357–368. Springer, Heidelberg (2005)Google Scholar
10. [10]
Nash-Williams, C.S.J.A.: On orientations, connectivity and odd vertex pairings in finite graphs. Canad. J. Math. 12, 555–567 (1960)
11. [11]
Parker, R., Rardin, R.: Guaranteed performance heuristics for the bottleneck traveling salesman problem. Oper. Res. Lett. 2(6), 269–272 (1984)
12. [12]
Robertson, N., Sanders, D., Seymour, P., Thomas, R.: The four-colour theorem. J. Comb. Theory Ser. B 70(1), 2–44 (1997)
13. [13]
Yi, S., Pei, Y., Kalyanaraman, S., Azimi-Sadjadi, B.: How is the capacity of ad hoc networks improved with directional antennas? Wireless Networks 13(5), 635–648 (2007)
14. [14]
Zhang, H., He, X.: On even triangulations of 2-connected embedded graphs. SIAM J. Comput. 34(3), 683–696 (2005)

© Springer-Verlag Berlin Heidelberg 2010

## Authors and Affiliations

• Evangelos Kranakis
• 1
• Oscar Morales Ponce
• 1