Navigating in a Graph by Aid of Its Spanning Tree

• Authors: Feodor F. Dragan;Martin Matamala
• Venue: ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
• Year: 2008

Let G = (V,E) be a graph andT be a spanning tree of G. We consider thefollowing strategy in advancing in G from a vertexx towards a target vertex y: from a currentvertex z (initially, z = x), unlessz = y, go to a neighbor of z inG that is closest to y in T (breakingties arbitrarily). In this strategy, each vertex has full knowledgeof its neighborhood in G and can use the distances inT to navigate in G. Thus, additionally tostandard local information (the neighborhood NG(v)), the only global informationthat is available to each vertex v is the topology of thespanning tree T (in fact, v can know only a verysmall piece of information about T and still be able toinfer from it the necessary tree-distances). For each source vertexx and target vertex y, this way, a path, called agreedy routing path, is produced. Denote by gG,T(x,y) the lengthof a longest greedy routing path that can be produced forx and y using this strategy and T. Wesay that a spanning tree T of a graph G is anadditive r -carcass for G if gG,T(x,y) ≤d G(x,y) +r for each ordered pair x,y εV. In this paper, we investigate the problem, given agraph family ${\cal F}$, whether a small integer r existssuch that any graph $G\in {\cal F}$ admits an additiver-carcass. We show that rectilinearp×q grids, hypercubes, distance-hereditarygraphs, dually chordal graphs (and, therefore, strongly chordalgraphs and interval graphs), all admit additive 0-carcasses.Furthermore, every chordal graph G admits an additive(ω(G) + 1)-carcass (whereω(G) is the size of a maximum clique ofG), each 3-sun-free chordal graph admits an additive2-carcass, each chordal bipartite graph admits an additive4-carcass. In particular, any k-tree admits an additive(k + 2)-carcass. All those carcasses are easy toconstruct.