Publication Title

SIAM Journal on Discrete Mathematics

Publication Date

2011

Document Type

Article

Keywords

navigating in graphs, spanning trees, distances, routing in graphs, hypercubes, rectilinear grids, chordal graphs, chordal bipartite graphs, dually chordal graphs, k-trees, efficient algorithms

Disciplines

Discrete Mathematics and Combinatorics

Abstract

Let $G=(V,E)$ be a graph and T be a spanning tree of G. We consider the following strategy in advancing in G from a vertex x towards a target vertex y: from a current vertex z (initially, $z=x$), unless $z=y$, go to a neighbor of z in G that is closest to y in T (breaking ties arbitrarily). In this strategy, each vertex has full knowledge of its neighborhood in G and can use the distances in T to navigate in G. Thus, additionally to standard local information (the neighborhood $N_G(v)$), the only global information that is available to each vertex v is the topology of the spanning tree T (in fact, v can know only a very small piece of information about T and still be able to infer from it the necessary tree-distances). For each source vertex x and target vertex y, this way, a path, called a greedy routing path, is produced. Denote by $g_{G,T}(x,y)$ the length of a longest greedy routing path that can be produced for x and y using this strategy and T. We say that a spanning tree T of a graph G is an additive r-carcass for G if $g_{G,T}(x,y)\leq d_G(x,y)+r$ for each ordered pair $x,y\in V$. In this paper, we investigate the problem, given a graph family $\mathcal{F}$, of whether a small integer r exists such that any graph $G\in\mathcal{F}$ admits an additive r-carcass. We show that rectilinear $p\times q$ grids, hypercubes, distance-hereditary graphs, dually chordal graphs (and, therefore, strongly chordal graphs and interval graphs) all admit additive 0-carcasses. Furthermore, every chordal graph G admits an additive $(\omega+1)$-carcass (where $\omega$ is the size of a maximum clique of G), each 3-sun-free chordal graph admits an additive 2-carcass, and each chordal bipartite graph admits an additive 4-carcass. In particular, any k-tree admits an additive $(k+2)$-carcass. All those carcasses are easy to construct in sequential as well as in distributed settings.

Comments

Copyright 2011 Society for Industrial and Applied Mathematics.


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