By David Peleg (auth.), Thomas Erlebach (eds.)

This booklet constitutes the completely refereed post-proceedings of the 3rd Workshop on Combinatorial and Algorithmic elements of Networking, held in Chester, united kingdom in July 2006, co-located with the thirteenth Colloquium on Structural details and communique Complexity, SIROCCO 2006.

The 10 revised complete papers including 1 invited lecture provided have been rigorously reviewed and chosen from a complete of twenty-two submissions. the themes lined diversity from the net graph to online game conception to thread matching, all within the context of large-scale networks.

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Additional info for Combinatorial and Algorithmic Aspects of Networking: Third Workshop, CAAN 2006, Chester, UK, July 2, 2006. Revised Papers

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Denote by vi,j the vertex at row i and column j, with 1 ≤ i ≤ n and 1 ≤ j ≤ m. For each vertex vi,j , the sum i + j may be even (even vertex) or odd (odd vertex). Each edge e of the grid is incident to one even vertex and one odd vertex. When replacing e with a path, the new vertices can be inserted closer to the even vertex than to the odd vertex. It is easy to see (see Fig. 3 (b)) that the LMST of the obtained set of vertices is G. In fact, dashed segments of Fig. 3 (b) are longer than the radius r, while dotted segments are not part of the LMST since for each one of them, say e, there is an axis-parallel alternative path whose segments are shorter than e.

Acknowledgments This article has its origins in the BICI - First Workshop on Graph Drawing, in Bertinoro, Italy, 2006. We thank Walter Didimo, Michael Kaufmann, and Giuseppe Liotta for organizing this workshop and posing the open problems. We also thank Franz Brandenburg and Sue Whitesides for stimulating discussions and observations on the LMST drawability problem. F. Cortese et al. References 1. P. Bose, P. Morin, I. Stojmenovic, and J. Urrutia. Routing with guaranteed delivery in ad hoc wireless networks.

Let d be the minimum distance between two vertices or between a vertex and an edge not incident on it. For each vertex u draw four circles d 2d 3d , 12 , 12 , and 4d ci , with 2 ≤ i ≤ 5, centered at that vertex, with radius 12 12 , respectively. Consider vertex u and suppose it has 5 incident edges (u, v1 ), (u, v2 ), (u, v3 ), (u, v4 ), and (u, v5 ), in this clockwise order around u. Edge (u, v1 ) remains unchanged. Draw four half-lines h2 , h3 , h4 , and h5 , starting from u with angles 72, 144, 216, and 288 degrees with respect to (u, v1 ), respectively, in this clockwise order around u.

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