### Navigating in a Graph by Aid of Its Spanning Tree Metric

01/01/2011Let $G=(V,E)$ 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.